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-\chapter{Irreducible Weight Modules}\label{ch:mathieu}
-
-\begin{definition}
- A representation \(V\) of \(\mathfrak{g}\) is called a \emph{weight
- \(\mathfrak{g}\)-module} if \(V = \bigoplus_{\lambda \in \mathfrak{h}^*}
- V_\lambda\) and \(\dim V_\lambda < \infty\) for all \(\lambda \in
- \mathfrak{h}^*\). The \emph{support of \(V\)} is the set
- \(\operatorname{supp} V = \{\lambda \in \mathfrak{h}^* : V_\lambda \ne 0\}\).
-\end{definition}
-
-\begin{example}
- Corollary~\ref{thm:finite-dim-is-weight-mod} is equivalent to the fact that
- every finite-dimensional representation of a semisimple Lie algebra is a
- weight module. More generally, every finite-dimensional irreducible
- representation of a reductive Lie algebra is a weight module.
-\end{example}
-
-\begin{example}\label{ex:submod-is-weight-mod}
- Proposition~\ref{thm:verma-is-weight-mod} and
- proposition~\ref{thm:max-verma-submod-is-weight} imply that the Verma module
- \(M(\lambda)\) and its maximal subrepresentation are both weight modules. In
- fact, the proof of proposition~\ref{thm:max-verma-submod-is-weight} is
- actually a proof of the fact that every subrepresentation \(W \subset V\) of
- a weight module \(V\) is a weight module, and \(W_\lambda = V_\lambda \cap
- W\) for all \(\lambda \in \mathfrak{h}^*\).
-\end{example}
-
-\begin{example}\label{ex:quotient-is-weight-mod}
- Given a weight module \(V\), a submodule \(W \subset V\) and \(\lambda \in
- \mathfrak{h}^*\), \(\left(\mfrac{V}{W}\right)_\lambda = \mfrac{V_\lambda}{W}
- \cong \mfrac{V_\lambda}{W_\lambda}\). In particular,
- \[
- \mfrac{V}{W}
- = \bigoplus_{\lambda \in \mathfrak{h}^*} \left(\mfrac{V}{W}\right)_\lambda
- \]
- is a weight module. It is clear that \(\mfrac{V_\lambda}{W} \subset
- \left(\mfrac{V}{W}\right)_\lambda\). To see that \(\mfrac{V_\lambda}{W} =
- \left(\mfrac{V}{W}\right)_\lambda\), we remark that \(V_\lambda \cong
- \mfrac{\mathcal{U}(\mathfrak{h})}{\mathfrak{m}_\lambda}
- \otimes_{\mathcal{U}(\mathfrak{h})} V\) as \(\mathfrak{h}\)-modules, where
- \(\mathfrak{m}_\lambda \normal \mathcal{U}(\mathfrak{h})\) is the left ideal
- generated by the elements \(H - \lambda(H)\), \(H \in \mathfrak{h}\).
- Likewise \(\left(\mfrac{V}{W}\right)_\lambda \cong
- \mfrac{\mathcal{U}(\mathfrak{h})}{\mathfrak{m}_\lambda}
- \otimes_{\mathcal{U}(\mathfrak{h})} \mfrac{V}{W}\) and the diagram
- \begin{center}
- \begin{tikzcd}
- V_\lambda \arrow{d} \arrow{r}{\pi} &
- \left(\mfrac{V}{W}\right)_\lambda \arrow{d} \\
- \mfrac{\mathcal{U}(\mathfrak{h})}{\mathfrak{m}_\lambda}
- \otimes_{\mathcal{U}(\mathfrak{h})} V
- \arrow[swap]{r}{\pi \otimes \operatorname{id}} &
- \mfrac{\mathcal{U}(\mathfrak{h})}{\mathfrak{m}_\lambda}
- \otimes_{\mathcal{U}(\mathfrak{h})} \mfrac{V}{W}
- \end{tikzcd}
- \end{center}
- commutes, so that the projection \(V_\lambda \to
- \left(\mfrac{V}{W}\right)_\lambda\) is surjective.
-\end{example}
-
-% TODO: Add an example of a module wich is NOT a weight module
-
-% TODOO: Prove this? Most likely not!
-\begin{proposition}\label{thm:centralizer-multiplicity}
- Let \(V\) be a completely reducible weight \(\mathfrak{g}\)-module. Then
- \(V_\lambda\) is a semisimple
- \(\mathcal{U}(\mathfrak{g})_0\)-module for any \(\lambda \in
- \mathfrak{h}^*\), where \(\mathcal{U}(\mathfrak{g})_0\) is
- the cetralizer of \(\mathfrak{h}\) in \(\mathcal{U}(\mathfrak{g})\).
- Moreover, the multiplicity of a given irreducible representation
- \(W\) of \(\mathfrak{g}\) coincides with the multiplicity of \(W_\lambda\) in
- \(V_\lambda\) as a \(\mathcal{U}(\mathfrak{g})_0\)-module,
- for any \(\lambda \in \operatorname{supp} V\).
-\end{proposition}
-
-\begin{definition}
- A weight \(\mathfrak{g}\)-module is called \emph{admissible} if \(\dim
- V_\lambda\) is bounded. The lowest upper bound for \(\dim V_\lambda\) is
- called \emph{the degree of \(V\)}. The \emph{essential support} of \(V\) is
- the set \(\operatorname{supp}_{\operatorname{ess}} V = \{ \lambda \in
- \mathfrak{h}^* : \dim V_\lambda = d \}\).
-\end{definition}
-
-\begin{example}\label{ex:laurent-polynomial-mod}
- There is a natural action of \(\mathfrak{sl}_2(K)\) in the space \(K[x,
- x^{-1}]\) of Laurent polynomials given by the formulas in
- (\ref{eq:laurent-polynomials-cusp-mod}). One can quickly verify \(K[x,
- x^{-1}]_{2 k} = K x^k\) and \(K[x, x^{-1}]_\lambda = 0\) for any \(\lambda
- \notin 2 \mathbb{Z}\), so that \(K[x, x^{-1}] = \bigoplus_{k \in \mathbb{Z}}
- K x^k\) is a degree \(1\) admissible weight \(\mathfrak{sl}_2(K)\)-module. It
- follows from example~\ref{ex:submod-is-weight-mod} that any non-zero
- subrepresentation \(W \subset K[x, x^{-1}]\) must contain a monomial \(x^k\).
- But since the operators \(-\frac{\mathrm{d}}{\mathrm{d}x} + \frac{x^{-1}}{2},
- x^2 \frac{\mathrm{d}}{\mathrm{d}x} + \frac{x}{2} : K[x, x^{-1}] \to K[x,
- x^{-1}]\) are both injective, this implies all other monomials can be found
- in \(W\) by successively applaying \(f\) and \(e\). Hence \(W = K[x,
- x^{-1}]\) and \(K[x, x^{-1}]\) is an irreducible representation.
- \begin{align}\label{eq:laurent-polynomials-cusp-mod}
- f \cdot p
- & = \left(- \frac{\mathrm{d}}{\mathrm{d}x} + \frac{x^{-1}}{2} \right) p &
- h \cdot p
- & = 2 x \frac{\mathrm{d}}{\mathrm{d}x} p &
- e \cdot p
- & = \left( x^2 \frac{\mathrm{d}}{\mathrm{d}x} + \frac{x}{2} \right) p
- \end{align}
-\end{example}
-
-% TODO: Point out supp_ess K[x^+-1] is 2Z, which is zariski dense
-% This proof is very technical, I don't think its worth including it
-\begin{proposition}
- Let \(V\) be an infinite-dimensional admissible representation of
- \(\mathfrak{g}\). The essential support
- \(\operatorname{supp}_{\operatorname{ess}} V\) is Zariski-dense in
- \(\mathfrak{h}^*\).
-\end{proposition}
-
-\begin{definition}
- A subalgebra \(\mathfrak{p} \subset \mathfrak{g}\) is called \emph{parabolic}
- if \(\mathfrak{b} \subset \mathfrak{p}\).
-\end{definition}
-
-% TODO: Comment afterwords that the Verma modules are indeed generalized Verma
-% modules
-\begin{definition}
- Given a parabolic subalgebra \(\mathfrak{p} \subset \mathfrak{g}\) and a
- \(\mathfrak{p}\)-module \(V\) the module \(M_{\mathfrak{p}}(V) =
- \operatorname{Ind}_{\mathfrak{p}}^{\mathfrak{g}} V\) is called \emph{a
- generalized Verma module}.
-\end{definition}
-
-\begin{proposition}
- Given an irreducible \(\mathfrak{p}\)-module \(V\), the generalized Verma
- module \(M_{\mathfrak{p}}(V)\) has a unique maximal subrepresentation
- \(N_{\mathfrak{p}}(V)\) and a unique irreducible quotient
- \(L_{\mathfrak{p}}(V) = \mfrac{M_{\mathfrak{p}}(V)}{N_{\mathfrak{p}}(V)}\).
- The irreducible quotient \(L_{\mathfrak{p}}(V)\) is a weight module.
-\end{proposition}
-
-\begin{definition}
- An irreducible \(\mathfrak{g}\)-module is called \emph{parabolic induced} if
- it is isomorphic to \(L_{\mathfrak{p}}(V)\) for some proper parabolic
- subalgebra \(\mathfrak{p} \subsetneq \mathfrak{g}\) and some
- \(\mathfrak{p}\)-module \(V\). An \emph{irreducible cuspidal
- \(\mathfrak{g}\)-module} is an irreducible representation of \(\mathfrak{g}\)
- which is \emph{not} parabolic induced.
-\end{definition}
-
-% TODO: Remark on the fact that any simple weight p-mod is a (p/u)-mod, so that
-% the notation of a cuspidal p-mod is well definited
-% TODO: Define the conjugation of a p-mod by an element of the Weil group?
-\begin{theorem}[Fernando]
- Any irreducible weight \(\mathfrak{g}\)-module is isomorphic to
- \(L_{\mathfrak{p}}(V)\) for some parabolic subalgebra \(\mathfrak{p} \subset
- \mathfrak{g}\) and some irreducible cuspidal \(\mathfrak{p}\)-module \(V\).
-\end{theorem}
-
-% TODO: Point out that the relationship between p-modules and cuspidal
-% g-modules is not 1-to-1
-
-\begin{proposition}[Fernando]
- Given a parabolic subalgebra \(\mathfrak{p} \subset \mathfrak{g}\), there
- exists a basis\footnote{This is usually called \emph{a $\mathfrak{p}$-adapted
- basis}} \(\Sigma\) for \(\Delta\) such that \(\Sigma \subset
- \Delta_{\mathfrak{p}_1}\). Furthermore, if \(\mathfrak{p}' \subset
- \mathfrak{g}\) is another parabolic subalgebra, \(V\) is an irreducible
- cuspidal \(\mathfrak{p}\)-module and \(W\) is an irreducible cuspidal
- \(\mathfrak{p}'\)-module then \(L_{\mathfrak{p}}(V) \cong
- L_{\mathfrak{p}'}(W)\) if, and only if \(\mathfrak{p}' = \mathfrak{p}^w\) and
- \(W \cong V^w\) for some \(w \in \mathcal{W}_V\), where
- \[
- \mathcal{W}_V
- = \langle
- T_\beta : \beta \in \Sigma, H_\beta + \mathfrak{u}
- \ \text{is central in}\ \mfrac{\mathfrak{p}}{\mathfrak{u}}
- \ \text{and}\ H_\beta\ \text{acts as a positive integer in}\ V
- \rangle
- \subset \mathcal{W}
- \]
-\end{proposition}
-
-% TODO: Point out that the definition of W_V is independant of the choice of
-% Sigma
-
-% TODO: Remark that the support of a simple weight module is always contained
-% in a coset
-\begin{corollary}[Fernando]\label{thm:cuspidal-mod-equivs}
- Let \(V\) be an irreducible weight \(\mathfrak{g}\)-module. The following
- conditions are equivalent.
- \begin{enumerate}
- \item \(V\) is cuspidal.
- \item \(F_\alpha\) acts injectively\footnote{This is what's usually refered
- to as a \emph{dense} representation in the literature.} in \(V\) for all
- \(\alpha \in \Delta\).
- \item The support of \(V\) is precisely one \(Q\)-coset\footnote{This is
- what's usually referred to as a \emph{torsion-free} representation in the
- literature.}.
- \end{enumerate}
-\end{corollary}
-
-\begin{example}
- As noted in example~\ref{ex:laurent-polynomial-mod}, the element \(f \in
- \mathfrak{sl}_2(K)\) acts injectively in the space of Laurent polynomials.
- Hence \(K[x, x^{-1}]\) is a cuspidal representation of
- \(\mathfrak{sl}_2(K)\).
-\end{example}
-
-% TODOO: Do we need this proposition? I think this only comes up in the
-% classification of simple completely reducible coherent families
-\begin{proposition}
- If \(\mathfrak{g} = \mathfrak{z} \oplus \mathfrak{s}_1 \oplus \cdots \oplus
- \mathfrak{s}_n\), where \(\mathfrak{z}\) is the center of \(\mathfrak{g}\)
- and \(\mathfrak{s}_i\) is a simple component of \(\mathfrak{g}\), then any
- irreducible weight \(\mathfrak{g}\)-module \(V\) decomposes as
- \[
- V = Z \otimes V_1 \otimes \cdots \otimes V_n
- \]
- where \(Z\) is a 1-dimensional representation of \(\mathfrak{z}\) and \(V_i\)
- is an irreducible weight \(\mathfrak{s}_i\)-module.
-\end{proposition}
-
-\begin{definition}
- A \emph{coherent family \(\mathcal{M}\) of degree \(d\)} is a weight
- \(\mathfrak{g}\)-module \(\mathcal{M}\) such that
- \begin{enumerate}
- \item \(\dim \mathcal{M}_\lambda = d\) for \emph{all} \(\lambda \in
- \mathfrak{h}^*\)
- \item For any \(u \in \mathcal{U}(\mathfrak{g})\) in the centralizer
- \(\mathcal{U}(\mathfrak{g})_0\) of \(\mathfrak{h}\) in
- \(\mathcal{U}(\mathfrak{g})\), the map
- \begin{align*}
- \mathfrak{h}^* & \to K \\
- \lambda & \mapsto
- \operatorname{Tr}(u\!\restriction_{\mathcal{M}_\lambda})
- \end{align*}
- is polynomial in \(\lambda\).
- \end{enumerate}
-\end{definition}
-
-% TODO: Add an example: there's an example of a coherent sl2-family in
-% Mathieu's paper
-% TODO: Add a discussion on how this may sound unintuitive, but the motivation
-% comes from the relationship between highest weight modules and coherent
-% families
-
-% TODO: Point out this is equivalent to M being a simple object in the
-% category of coherent families
-\begin{definition}
- A coherent family \(\mathcal{M}\) is called \emph{irreducible} if
- \(\mathcal{M}_\lambda\) is a simple
- \(\mathcal{U}(\mathfrak{g})_0\)-module for some \(\lambda \in
- \mathfrak{h}^*\).
-\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}
-
-% 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
-\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,
- \[
- \mathcal{M}^{\operatorname{ss}}
- \cong \bigoplus_{i j} \mfrac{\mathcal{M}_{i j + 1}}{\mathcal{M}_{i j}}
- \]
-\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
- \[
- \mathcal{M}^{\operatorname{ss}}[\lambda_i]
- \cong \bigoplus_j \mfrac{\mathcal{M}_{i j + 1}}{\mathcal{M}_{i j}}
- \]
-
- As for the existence of the semisimplification, it suffices to show
- \[
- \mathcal{M}^{\operatorname{ss}}
- = \bigoplus_{i j} \mfrac{\mathcal{M}_{i j + 1}}{\mathcal{M}_{i j}}
- \]
- 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
- \(\mu \in \lambda_k + Q\)
- \[
- \mathcal{M}_\mu^{\operatorname{ss}}
- = \bigoplus_{i j}
- \left(
- \mfrac{\mathcal{M}_{i j + 1}}{\mathcal{M}_{i j}}
- \right)_\mu
- = \bigoplus_j
- \left(
- \mfrac{\mathcal{M}_{k j + 1}}{\mathcal{M}_{k j}}
- \right)_\mu
- \cong \bigoplus_j
- \mfrac{(\mathcal{M}_{k j + 1})_\mu}
- {(\mathcal{M}_{k j})_\mu}
- \]
-
- In particular,
- \[
- \dim \mathcal{M}_\mu^{\operatorname{ss}}
- = \sum_j
- \dim (\mathcal{M}_{k j + 1})_\mu
- - \dim (\mathcal{M}_{k j})_\mu
- = \dim \mathcal{M}[\lambda_k]_\mu
- = \dim \mathcal{M}_\mu
- = d
- \]
-
- Likewise, given \(u \in \mathcal{U}(\mathfrak{g})_0\) the value
- \[
- \operatorname{Tr}(u\!\restriction_{\mathcal{M}_\mu^{\operatorname{ss}}})
- = \sum_j
- \operatorname{Tr}(u\!\restriction_{(\mathcal{M}_{k j + 1})_\mu})
- - \operatorname{Tr}(u\!\restriction_{(\mathcal{M}_{k j})_\mu})
- = \operatorname{Tr}(u\!\restriction_{\mathcal{M}[\lambda_k]_\mu})
- = \operatorname{Tr}(u\!\restriction_{\mathcal{M}_\mu})
- \]
- is polynomial in \(\mu \in \mathfrak{h}^*\).
-\end{proof}
-
-\begin{corollary}\label{thm:admissible-is-submod-of-extension}
- 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{lemma}
- Let \(\mathcal{M}\) be a coherent family. The set \(U = \{\lambda \in
- \mathfrak{h}^* : \mathcal{M}_\lambda \ \text{is a simple
- $\mathcal{U}(\mathfrak{g})_0$-module}\}\) is Zariski-open.
-\end{lemma}
-
-\begin{proof}
- For each \(\lambda \in \mathfrak{h}^*\) we introduce the bilinear form
- \begin{align*}
- B_\lambda : \mathcal{U}(\mathfrak{g})_0 \times \mathcal{U}(\mathfrak{g})_0
- & \to K \\
- (u, v)
- & \mapsto \operatorname{Tr}(u v \!\restriction_{\mathcal{M}_\lambda})
- \end{align*}
- and consider its rank -- i.e. the dimension of the image of the induced
- operator
- \begin{align*}
- \mathcal{U}(\mathfrak{g})_0 & \to \mathcal{U}(\mathfrak{g})_0^* \\
- u & \mapsto B_\lambda(u, \cdot)
- \end{align*}
-
- Our first observation is that \(\operatorname{rank} B_\lambda \le d^2\). This
- follows from the commutativity of
- \begin{center}
- \begin{tikzcd}
- \mathcal{U}(\mathfrak{g})_0 \arrow{r} \arrow{d} &
- \mathcal{U}(\mathfrak{g})_0^* \\
- \operatorname{End}(\mathcal{M}_\lambda) \arrow{r}{\sim} &
- \operatorname{End}(\mathcal{M}_\lambda)^* \arrow{u}
- \end{tikzcd},
- \end{center}
- where the map \(\mathcal{U}(\mathfrak{g})_0 \to
- \operatorname{End}(\mathcal{M}_\lambda)\) is given by the action of
- \(\mathcal{U}(\mathfrak{g})_0\), the map
- \(\operatorname{End}(\mathcal{M}_\lambda)^* \to
- \mathcal{U}(\mathfrak{g})_0^*\) is its dual, and the isomorphism
- \(\operatorname{End}(\mathcal{M}_\lambda) \isoto
- \operatorname{End}(\mathcal{M}_\lambda)^*\) is induced by the trace form
- \begin{align*}
- \operatorname{End}(\mathcal{M}_\lambda) \times
- \operatorname{End}(\mathcal{M}_\lambda) & \to K \\
- (T, S) & \mapsto \operatorname{Tr}(T S)
- \end{align*}
-
- Indeed, \(\operatorname{rank} B_\lambda \le
- \operatorname{rank}(\mathcal{U}(\mathfrak{g})_0 \to
- \operatorname{End}(\mathcal{M}_\lambda)) \le \dim
- \operatorname{End}(\mathcal{M}_\lambda) = d^2\). Furthermore, if
- \(\operatorname{rank} B_\lambda = d^2\) then we must have
- \(\operatorname{rank}(\mathcal{U}(\mathfrak{g})_0 \to
- \operatorname{End}(\mathcal{M}_\lambda)) = d^2\) -- i.e. the map
- \(\mathcal{U}(\mathfrak{g})_0 \to \operatorname{End}(\mathcal{M}_\lambda)\)
- is surjective. In particular, if \(\operatorname{rank} B_\lambda = d^2\) then
- \(\mathcal{M}_\lambda\) is a simple \(\mathcal{U}(\mathfrak{g})_0\)-module,
- for if \(V \subset \mathcal{M}_\lambda\) is invariant under the action of
- \(\mathcal{U}(\mathfrak{g})_0\) then \(V\) is invariant under any
- \(K\)-linear operator \(\mathcal{M}_\lambda \to \mathcal{M}_\lambda\), so
- that \(W = 0\) or \(W = \mathcal{M}_\lambda\).
-
- On the other hand, if \(\mathcal{M}_\lambda\) is simple then by Burnside's
- theorem on matrix algebras the map \(\mathcal{U}(\mathfrak{g})_0 \to
- \operatorname{End}(\mathcal{M}_\lambda)\) is surjective. Hence the
- commutativity of the previously drawn diagram, as well as the fact that
- \(\operatorname{rank}(\mathcal{U}(\mathfrak{g})_0 \to
- \operatorname{End}(\mathcal{M}_\lambda)) =
- \operatorname{rank}(\operatorname{End}(\mathcal{M}_\lambda)^* \to
- \mathcal{U}(\mathfrak{g})_0^*)\), imply that \(\operatorname{rank} B_\lambda
- = d^2\). This goes to show that \(U\) is precisely the set of \(\lambda\)
- such that \(B_\lambda\) has maximal rank \(d^2\). We now show that \(U\) is
- Zariski-open. First, notice that
- \[
- U =
- \bigcup_{\substack{W \subset \mathcal{U}(\mathfrak{g})_0 \\ \dim W = d^2}}
- U_W,
- \]
- where \(U_W = \{\lambda \in \mathcal{U}(\mathfrak{g})_0 : \operatorname{rank}
- B_\lambda\!\restriction_W = d^2 \}\).
-
- Indeed, if \(\operatorname{rank} B_\lambda = d^2\) it follows from the
- surjectivity of the map \(\mathcal{U}(\mathfrak{g})_0 \to
- \operatorname{End}(\mathcal{M}_\lambda)\) that there is some \(W \subset
- \mathcal{U}(\mathfrak{g})_0\) with \(\dim W = d^2\) such that the restriction
- \(W \to \operatorname{End}(\mathcal{M}_\lambda)\) is surjective. The
- comutativity of
- \begin{center}
- \begin{tikzcd}
- W \arrow{r} \arrow{d} & W^* \\
- \operatorname{End}(\mathcal{M}_\lambda) \arrow{r}{\sim} &
- \operatorname{End}(\mathcal{M}_\lambda)^* \arrow{u}
- \end{tikzcd}
- \end{center}
- then implies \(\operatorname{rank} B_\lambda\!\restriction_W = d^2\). In
- other words, \(U \subset \bigcup_W U_W\).
-
- Likewise, if \(\operatorname{rank} B_\lambda\!\restriction_W = d^2\) for some
- \(W\), then the commutativity of
- \begin{center}
- \begin{tikzcd}
- W \arrow{r} \arrow{d} & W^* \\
- \mathcal{U}(\mathfrak{g})_0 \arrow{r} &
- \mathcal{U}(\mathfrak{g})_0^* \arrow{u}
- \end{tikzcd}
- \end{center}
- implies \(\operatorname{rank} B_\lambda \ge d^2\), which goes to show
- \(\bigcup_W U_W \subset U\).
-
- Given \(\lambda \in U_W\), the surjectivity of \(W \to
- \operatorname{End}(\mathcal{M}_\lambda)\) and the fact that \(\dim W <
- \infty\) imply \(W \to W^*\) is invertible. Since \(\mathcal{M}\) is a
- coherent family, \(B_\lambda\) depends polynomialy in \(\lambda\). Hence so
- does the induced maps \(W \to W^*\). In particular, there is some Zariski
- neighborhood \(V\) of \(\lambda\) such that the map \(W \to W^*\) induced by
- \(B_\mu\!\restriction_W\) is invertible for all \(\mu \in V\).
-
- But the surjectivity of the map induced by \(B_\mu\!\restriction_W\) implies
- \(\operatorname{rank} B_\mu = d^2\), so \(\mu \in U_W\) and therefore \(V
- \subset U_W\). This implies \(U_W\) is open for all \(W\). Finally, \(U\) is
- the union of Zariski-open subsets and is therefore open. We are done.
-\end{proof}
-
-\begin{theorem}[Mathieu]
- Let \(\mathcal{M}\) be an irreducible 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)}.
-
- 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 irreducible 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
-% given eigenspace by translating by the F's and E's, but neither of those are
-% injective in general, so the translation could take nonzero vectors to zero.
-% If the F's were invertible this problem wouldn't exist, so we might as well
-% invert them by force!
-
-\begin{definition}
- Let \(R\) be a ring. A subset \(S \subset R\) is called \emph{multiplicative}
- if \(s \cdot t \in S\) for all \(s, t \in S\) and \(0 \notin S\). A
- multiplicative subset \(S\) is said to satisfy \emph{Ore's localization
- condition} if for each \(r \in R\), \(s \in S\) there exists \(u_1, u_2 \in
- R\) and \(t_1, t_2 \in S\) such that \(s r = u_1 t_1\) and \(r s = t_2 u_2\).
-\end{definition}
-
-\begin{theorem}[Ore-Asano]
- Let \(S \subset R\) be a multiplicative subset satisfying Ore's localization
- condition. Then there exists a (unique) ring \(S^{-1} R\), with a canonical
- ring homomorphism \(R \to S^{-1} R\), and enjoying the universal property
- that each ring homomorphism \(f : R \to T\) such that \(f(s)\) is invertible
- for all \(s \in S\) can be uniquely extended to a ring homomorphism \(S^{-1}
- R \to T\). \(S^{-1} R\) is called \emph{the localization of \(R\) by \(S\)},
- and the map \(R \to S^{-1} R\) is called \emph{the localization map}.
- \begin{center}
- \begin{tikzcd}
- S^{-1} R \arrow[dotted]{rd} & \\
- R \arrow{u} \arrow[swap]{r}{f} & T
- \end{tikzcd}
- \end{center}
-\end{theorem}
-
-% TODO: Cite the discussion of goodearl-warfield, chap 6, on how to derive the
-% localization condition
-% TODO: In general checking that a set satisfies Ore's condition can be tricky,
-% but there is an easyer condition given by the lemma
-
-\begin{lemma}
- Let \(S \subset R\) be a multiplicative subset generated by locally
- \(\operatorname{ad}\)-nilpotent elements -- i.e. elements \(s \in S\) such
- that for each \(r \in R\) there exists \(n > 0\) such that \(\operatorname{ad}(s)^n r = [s, [s, \cdots
- [s, r]]\cdots] = 0\). Then \(S\) satisfies Ore's
- localization condition.
-\end{lemma}
-
-\begin{definition}
- Let \(S \subset R\) be a multiplicative subset satisfying Ore's localization
- condition and \(M\) be a \(R\)-module. The \(S^{-1} R\)-module \(S^{-1} M =
- S^{-1} R \otimes_R M\) is called \emph{the localization of \(M\) by \(S\)},
- and the homomorphism of \(R\)-modules
- \begin{align*}
- M & \to S^{-1} M \\
- m & \mapsto 1 \otimes m
- \end{align*}
- is called \emph{the localization map of \(M\)}.
-\end{definition}
-
-% TODO: Point out that the localization of modules is functorial
-
-% TODO: Point out the the localization map is in not injective in general
-\begin{lemma}
- Let \(S \subset R\) be a multiplicative subset satisfying Ore's localization
- condition and \(M\) be a \(R\)-module. If \(S\) acts injectively in \(M\)
- then the localization map \(M \to S^{-1} M\) is injective. In particular, if
- \(S\) has no zero divisors then \(R\) is a subring of \(S^{-1} R\).
-\end{lemma}
-
-% TODO: Point out that S^-1 M can be seen as a R-module, where R acts via the
-% localization map
-% TODO: Point out that each element of the localization has the form s^-1 r
-
-% TODO: Point out that Sigma depends on V!
-\begin{lemma}\label{thm:nice-basis-for-inversion}
- Let \(V\) be an irreducible infinite-dimensional admissible
- \(\mathfrak{g}\)-module. There is a basis \(\Sigma = \{\beta_1, \ldots,
- \beta_n\}\) of \(\Delta\) such that the elements \(F_{\beta_i}\) all act
- injectively on \(V\) and satisfy \([F_{\beta_i}, F_{\beta_j}] = 0\).
-\end{lemma}
-
-\begin{corollary}
- Let \(\Sigma\) be as in lemma~\ref{thm:nice-basis-for-inversion} and
- \((F_\beta)_{\beta \in \Sigma} \subset \mathcal{U}(\mathfrak{g})\) be the
- multiplicative subset generated by the \(F_\beta\)'s. The \(K\)-algebra
- \(\Sigma^{-1} \mathcal{U}(\mathfrak{g}) = (F_\beta)_{\beta \in \Sigma}^{-1}
- \mathcal{U}(\mathfrak{g})\) is well defined. Moreover, if we denote by
- \(\Sigma^{-1} V\) the localization of \(V\) by \((F_\beta)_{\beta \in
- \Sigma}\), the localization map \(V \to \Sigma^{-1} V\) is injective.
-\end{corollary}
-
-% TODO: Fix V and Sigma beforehand
-\begin{proposition}\label{thm:irr-admissible-is-contained-in-nice-mod}
- The the restriction of the localization \(\Sigma^{-1} V\) is an admissible
- \(\mathfrak{g}\)-module of degree \(d\) with \(\operatorname{supp}
- \Sigma^{-1} V = Q + \operatorname{supp} V\) and \(\dim \Sigma^{-1} V_\lambda
- = d\) for all \(\lambda \in \operatorname{supp} \Sigma^{-1} V\).
-\end{proposition}
-
-\begin{proof}
- Fix some \(\beta \in \Sigma\). We begin by showing that \(F_\beta\) and
- \(F_\beta^{-1}\) map the weight space \(\Sigma^{-1} V_\lambda\) to the weight
- spaces \(\Sigma^{-1} V_{\lambda - \beta}\) and \(\Sigma^{-1} V_{\lambda +
- \beta}\) respectively. Indeed, given \(v \in V_\lambda\) and \(H \in
- \mathfrak{h}\) we have
- \[
- H F_\beta v
- = ([H, F_\beta] + F_\beta H)v
- = F_\beta (-\beta(H) + H) v
- = F_\beta (\lambda - \beta)(H) \cdot v
- = (\lambda - \beta)(H) \cdot F_\beta v
- \]
-
- On the other hand,
- \[
- 0
- = [H, 1]
- = [H, F_\beta F_\beta^{-1}]
- = F_\beta [H, F_\beta^{-1}] + [H, F_\beta] F_\beta^{-1}
- = F_\beta [H, F_\beta^{-1}] - \beta(H) F_\beta F_\beta^{-1},
- \]
- so that \([H, F_\beta^{-1}] = \beta(H) \cdot F_\beta^{-1}\) and therefore
- \[
- H F_\beta^{-1} v
- = ([H, F_\beta^{-1}] + F_\beta^{-1} H) v
- = F_\beta^{-1} (\beta(H) + H) v
- = (\lambda + \beta)(H) \cdot F_\beta^{-1} v
- \]
-
- From the fact that \(F_\beta^{\pm 1}\) maps \(V_\lambda\) to \(\Sigma^{-1}
- V_{\lambda \pm \beta}\) follows our first conclusion: since \(V\) is a weight
- module and every element of \(\Sigma^{-1} V\) has the form \(s^{-1} v =
- s^{-1} \otimes v\) for \(s \in (F_\beta)_{\beta \in \Sigma}\) and \(v \in
- V\), we can see that \(\Sigma^{-1} V = \bigoplus_\lambda \Sigma^{-1}
- V_\lambda\). Furtheremore, since the action of each \(F_\beta\) in
- \(\Sigma^{-1} V\) is bijective and \(\Sigma\) is a basis of \(Q\) we obtain
- \(\operatorname{supp} \Sigma^{-1} V = Q + \operatorname{supp} V\).
-
- Again, because of the bijectivity of the \(F_\beta\)'s, to see that \(\dim
- \Sigma^{-1} V_\lambda = d\) for all \(\lambda \in \operatorname{supp}
- \Sigma^{-1} V\) it suffices to show that \(\dim \Sigma^{-1} V_\lambda = d\)
- for some \(\lambda \in \operatorname{supp} \Sigma^{-1} V\). We may take
- \(\lambda \in \operatorname{supp} V\) with \(\dim V_\lambda = d\). For any
- finite-dimensional subspace \(W \subset \Sigma^{-1} V_\lambda\) we can find
- \(s \in (F_\beta)_{\beta \in \Sigma}\) such that \(s W \subset V\). If \(s =
- F_{\beta_{i_1}} \cdots F_{\beta_{i_n}}\), it is clear \(s W \subset
- V_{\lambda - \beta_{i_1} - \cdots - \beta_{i_n}}\), so \(\dim W = \dim sW \le
- d\). This holds for all finite-dimensional \(W \subset \Sigma^{-1}
- V_\lambda\), so \(\dim \Sigma^{-1} V_\lambda \le d\). It then follows from
- the fact that \(V_\lambda \subset \Sigma^{-1} V_\lambda\) that \(V_\lambda =
- \Sigma^{-1} V_\lambda\) and therefore \(\dim \Sigma^{-1} V_\lambda = d\).
-\end{proof}
-
-\begin{proposition}\label{thm:nice-automorphisms-exist}
- There is a family of automorphisms \(\{\theta_\lambda : \Sigma^{-1}
- \mathcal{U}(\mathfrak{g}) \to \Sigma^{-1}
- \mathcal{U}(\mathfrak{g})\}_{\lambda \in \mathfrak{h}^*}\) such that
- \begin{enumerate}
- \item \(\theta_{k_1 \beta_1 + \cdots + k_n \beta_n}(r) = F_{\beta_1}^{k_1}
- \cdots F_{\beta_n}^{k_n} r F_{\beta_1}^{- k_n} \cdots F_{\beta_n}^{-
- k_1}\) for all \(r \in \Sigma^{-1} \mathcal{U}(\mathfrak{g})\) and \(k_1,
- \ldots, k_n \in \mathbb{Z}\).
-
- \item For each \(r \in \Sigma^{-1} \mathcal{U}(\mathfrak{g})\) the map
- \begin{align*}
- \mathfrak{h}^* & \to \Sigma^{-1} \mathcal{U}(\mathfrak{g}) \\
- \lambda & \mapsto \theta_\lambda(r)
- \end{align*}
- is polynomial.
-
- \item If \(\lambda, \mu \in \mathfrak{h}^*\), \(M\) is a \(\Sigma^{-1}
- \mathcal{U}(\mathfrak{g})\)-module whose restriction to
- \(\mathcal{U}(\mathfrak{g})\) is a weight \(\mathfrak{g}\)-module and
- \(\theta_\lambda M\) is the \(\Sigma^{-1}
- \mathcal{U}(\mathfrak{g})\)-module \(M\) twisted by the automorphism
- \(\theta_\lambda\) then \(M_\mu = (\theta_\lambda M)_{\mu + \lambda}\).
- In particular, \(\operatorname{supp} \theta_\lambda M = \lambda +
- \operatorname{supp} M\).
- \end{enumerate}
-\end{proposition}
-
-\begin{proof}
- Since the elements \(F_\beta\), \(\beta \in \Sigma\) commute with one
- another, the endomorphisms
- \begin{align*}
- \theta_{k_1 \beta_1 + \cdots + k_n \beta_n}
- : \Sigma^{-1} \mathcal{U}(\mathfrak{g}) &
- \to \Sigma^{-1} \mathcal{U}(\mathfrak{g}) \\
- r & \mapsto
- F_{\beta_1}^{k_1} \cdots F_{\beta_n}^{k_n}
- r
- F_{\beta_1}^{- k_n} \cdots F_{\beta_n}^{- k_1}
- \end{align*}
- are well defined for all \(k_1, \ldots, k_n \in \mathbb{Z}\).
-
- Fix some \(r \in \Sigma^{-1} \mathcal{U}(\mathfrak{g})\). For any \(s \in
- (F_\beta)_{\beta \in \Sigma}\) and \(k > 0\) we have \(s^k r = \binom{k}{0}
- \operatorname{ad}(s)^0 r s^{k - 0} + \cdots + \binom{k}{k}
- \operatorname{ad}(s)^k r s^{k - k}\). Now if we take \(m\) such
- \(\operatorname{ad}(F_\beta)^{m + 1} r = 0\) for all \(\beta \in \Sigma\) we
- find
- \[
- \theta_{k_1 \beta_1 + \cdots + k_n \beta_n}(r)
- = \sum_{i_1, \ldots, i_n = 1, \ldots, m}
- \binom{k_1}{i_1} \cdots \binom{k_n}{i_n}
- \operatorname{ad}(F_{\beta_1})^{i_1} \cdots
- \operatorname{ad}(F_{\beta_n})^{i_n}
- r
- F_{\beta_1}^{- i_1} \cdots F_{\beta_n}^{- i_n}
- \]
- for all \(k_1, \ldots, k_n \in \NN\).
-
- Since the binomial coeffients \(\binom{x}{k} = \frac{x (x -1) \cdots (x - k +
- 1)}{k!}\) can be uniquely extended to polynomial functions in \(x \in K\), we
- may in general define
- \[
- \theta_\lambda(r)
- = \sum_{i_1, \ldots, i_n \ge 0}
- \binom{\lambda_1}{i_1} \cdots \binom{\lambda_n}{i_n}
- \operatorname{ad}(F_{\beta_1})^{i_1} \cdots
- \operatorname{ad}(F_{\beta_n})^{i_n}
- r
- F_{\beta_1}^{- i_1} \cdots F_{\beta_n}^{- i_n}
- \]
- for \(\lambda_1, \ldots, \lambda_n \in K\), \(\lambda = \lambda_1 \beta_1 +
- \cdots + \lambda_n \beta_n \in \mathfrak{h}^*\)
-
- It is clear that the \(\theta_\lambda\) are endmorphisms. To see that the
- \(\theta_\lambda\) are indeed automorphisms, notice \(\theta_{- k_1 \beta_1 -
- \cdots - k_n \beta_n} = \theta_{k_1 \beta_1 + \cdots + k_n \beta_n}^{-1}\).
- The uniqueness of the polynomial extensions then implies \(\theta_{- \lambda}
- = \theta_\lambda^{-1}\) in general: given \(r \in \Sigma^{-1}
- \mathcal{U}(\mathfrak{g})\), the map
- \begin{align*}
- \mathfrak{h}^* & \to \Sigma^{-1} \mathcal{U}(\mathfrak{g}) \\
- \lambda & \mapsto \theta_\lambda(\theta_{-\lambda}(r)) - r
- \end{align*}
- is a polynomial extension of the zero map \(\ZZ \beta_1 \oplus \cdots \oplus
- \ZZ \beta_n \to \Sigma^{-1} \mathcal{U}(\mathfrak{g})\) and is therefore
- identicaly zero.
-
- Finally, let \(M\) be a \(\Sigma^{-1} \mathcal{U}(\mathfrak{g})\)-module
- whose restriction is a weight module. If \(m \in M\) then
- \[
- m \in (\theta_\lambda M)_{\mu + \lambda}
- \iff \theta_\lambda(H) m = (\mu + \lambda)(H) \cdot m
- \, \forall H \in \mathfrak{h}
- \]
-
- But
- \[
- \theta_\beta(H)
- = F_\beta H F_\beta^{-1}
- = ([F_\beta, H] + H F_\beta) F_\beta^{-1}
- = (\beta(H) + H) F_\beta F_\beta^{-1}
- = \beta(H) + H
- \]
- for all \(H \in \mathfrak{h}\) and \(\beta \in \Sigma\). In general,
- \(\theta_\lambda(H) = \lambda(H) + H\) for all \(\lambda \in \mathfrak{h}^*\)
- and hence
- \[
- \begin{split}
- m \in (\theta_\lambda M)_{\mu + \lambda}
- & \iff (\lambda(H) + H) m = (\mu + \lambda)(H) \cdot m
- \; \forall H \in \mathfrak{h} \\
- & \iff H m = \mu(H) \cdot m \; \forall H \in \mathfrak{h} \\
- & \iff m \in M_\mu
- \end{split},
- \]
- so that \((\theta_\lambda M)_{\mu + \lambda} = M_\mu\).
-\end{proof}
-
-\begin{proposition}[Mathieu]
- There exists a coherent extension \(\mathcal{M}\) of \(V\).
-\end{proposition}
-
-\begin{proof}
- Let \(\Lambda\) be a set of representatives of the \(Q\)-cosets in
- \(\mathfrak{h}^*\) with \(0 \in \Lambda\) and take
- \[
- \mathcal{M}
- = \bigoplus_{\lambda \in \Lambda} \theta_\lambda \Sigma^{-1} V
- \]
-
- On the one hand, \(V\) lies in \(\Sigma^{-1} V = \theta_0 \Sigma^{-1} V\) --
- notice that \(\theta_0\) is just the identity operator -- and therefore \(V
- \subset \mathcal{M}\). On the other hand, \(\dim \mathcal{M}_\mu = \dim
- \theta_\lambda \Sigma^{-1} V_\mu = \dim \Sigma^{-1} V_{\mu - \lambda} = d\)
- for all \(\mu \in \lambda + Q\), \(\lambda \in \Lambda\). Furtheremore, given
- \(u \in \mathcal{U}(\mathfrak{g})_0\) and \(\mu \in \lambda +
- Q\),
- \[
- \operatorname{Tr}(u\!\restriction_{\mathcal{M}_\mu})
- = \operatorname{Tr}
- (\theta_\lambda(u)\!\restriction_{\Sigma^{-1} V_{\mu - \lambda}})
- \]
- is polynomial in \(\mu\) because of the second item of
- proposition~\ref{thm:nice-automorphisms-exist}.
-\end{proof}
-
-\begin{theorem}[Mathieu]
- 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, \(\operatorname{Ext}(V)\) is
- irreducible as a coherent family.
-\end{theorem}
-
-\begin{proof}
- The existence part should be clear from the previous discussion: it suffices
- to fix some coherent extension \(\mathcal{M}\) of \(V\) and take
- \(\operatorname{Ext}(V) = \mathcal{M}^{\operatorname{ss}}\).
-
- 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 proposition~\ref{thm:centralizer-multiplicity}
- that \(\operatorname{Ext}(V)_\lambda = V_\lambda\) is a simple
- \(\mathcal{U}(\mathfrak{g})_0\)-module for some \(\lambda \in
- \operatorname{supp} V\).
-
- As for the uniqueness of \(\operatorname{Ext}(V)\), fix some other completely
- reducible coherent extension \(\mathcal{N}\) of \(V\). We claim that the
- multiplicity of a given irreducible \(\mathfrak{g}\)-module \(W\) in
- \(\mathcal{N}\) is determined by its \emph{trace function}
- \begin{align*}
- \mathfrak{h}^* \times \mathcal{U}(\mathfrak{g})_0 &
- \to K \\
- (\lambda, u) &
- \mapsto \operatorname{Tr}(u\!\restriction_{\mathcal{N}_\lambda})
- \end{align*}
-
- % TODO: Point out that this multiplicity is determined by the characters
- % beforehand
- Indeed, given \(\lambda \in \operatorname{supp} V\) the multiplicity of \(W\)
- in \(\mathcal{N}\) is the same as the multiplicity of \(W_\lambda\) in
- \(\mathcal{N}_\lambda\), which is determined by the character
- \(\chi_{\mathcal{N}_\lambda} : \mathcal{U}(\mathfrak{g})_0
- \to K\) -- see proposition~\ref{thm:centralizer-multiplicity}. We now claim
- that the trace function of \(\mathcal{N}\) is the same as that of
- \(\operatorname{Ext}(V)\). Clearly,
- \(\operatorname{Tr}(u\!\restriction_{\operatorname{Ext}(V)_\lambda})
- = \operatorname{Tr}(u\!\restriction_{V_\lambda})
- = \operatorname{Tr}(u\!\restriction_{\mathcal{N}_\lambda})\) for all
- \(\lambda \in \operatorname{supp}_{\operatorname{ess}} V\), \(u \in
- \mathcal{U}(\mathfrak{g})_0\). Since the essential support of
- \(V\) is Zariski-dense and the maps \(\lambda \mapsto
- \operatorname{Tr}(u\!\restriction_{\operatorname{Ext}(V)_\lambda})\) and
- \(\lambda \mapsto \operatorname{Tr}(u\!\restriction_{\mathcal{N}_\lambda})\)
- are polynomial in \(\lambda \in \mathfrak{h}^*\), it follows that this maps
- coincide for all \(u\).
-
- In conclusion, \(\mathcal{N} \cong \operatorname{Ext}(V)\) and
- \(\operatorname{Ext}(V)\) is unique.
-\end{proof}
-
-\begin{proposition}[Mathieu]
- The central characters of the irreducible submodules of
- \(\operatorname{Ext}(V)\) are all the same.
-\end{proposition}