diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -29,25 +29,23 @@ the regular \(\mathfrak{g}\)-module then \(\mathcal{U}(\mathfrak{g})_\lambda =
= 0
\subsetneq \mathcal{U}(\mathfrak{g})
\]
-and the weight space decomposition -- i.e.
-corollary~\ref{thm:finite-dim-is-weight-mod} -- fails for
-\(\mathcal{U}(\mathfrak{g})\).
-
-Indeed, our proof of corollary~\ref{thm:finite-dim-is-weight-mod} relied
-heavily in the simultaneous diagonalization of commuting operators in a
-finite-dimensional space. Even if we restrict ourselves to irreducible modules,
-there is still a diverse spectrum of counterexamples to
+and the weight space decomposition fails for \(\mathcal{U}(\mathfrak{g})\).
+
+Indeed, our proof of the weight space decomposition in the finite-dimensional
+case relied heavily in the simultaneous diagonalization of commuting operators
+in a finite-dimensional space. Even if we restrict ourselves to irreducible
+modules, there is still a diverse spectrum of counterexamples to
corollary~\ref{thm:finite-dim-is-weight-mod} in the infinite-dimensional
setting. For instance, any representation \(V\) of \(\mathfrak{g}\) whose
restriction to \(\mathfrak{h}\) is a free module satisfies \(V_\lambda = 0\)
for all \(\lambda\) as in the previous example. These are called
-\(\mathfrak{h}\)-free representations, and rank \(1\) irreducible
+\emph{\(\mathfrak{h}\)-free representations}, and rank \(1\) irreducible
\(\mathfrak{h}\)-free \(\mathfrak{sp}_{2 n}(K)\)-modules where first classified
by Nilsson in \cite{nilsson}. Dimitar's construction of the so called
\emph{exponential tensor \(\mathfrak{sl}_n(K)\)-modules} in \cite{dimitar-exp}
is also an interesting source of counterexamples.
-Since the weight spaces decomposition was perhaps the single most instrumental
+Since the weight space decomposition was perhaps the single most instrumental
ingredient of our previous analysis, it is only natural to restrict ourselves
to the case it holds. This brings us to the following definition.
@@ -87,21 +85,21 @@ to the case it holds. This brings us to the following definition.
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 \(V_\lambda \cong
- \mfrac{\mathcal{U}(\mathfrak{h})}{\mathfrak{m}_\lambda}
+ \mfrac{\mathcal{U}(\mathfrak{h})}{I_\lambda}
\otimes_{\mathcal{U}(\mathfrak{h})} V\) as \(\mathfrak{h}\)-modules, where
- \(\mathfrak{m}_\lambda \normal \mathcal{U}(\mathfrak{h})\) is the left ideal
+ \(I_\lambda \normal \mathcal{U}(\mathfrak{h})\) is the left ideal
generated by the elements \(H - \lambda(H)\), \(H \in \mathfrak{h}\).
Likewise \(\left(\mfrac{V}{W}\right)_\lambda \cong
- \mfrac{\mathcal{U}(\mathfrak{h})}{\mathfrak{m}_\lambda}
+ \mfrac{\mathcal{U}(\mathfrak{h})}{I_\lambda}
\otimes_{\mathcal{U}(\mathfrak{h})} \mfrac{V}{W}\) 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}
+ \mfrac{\mathcal{U}(\mathfrak{h})}{I_\lambda}
\otimes_{\mathcal{U}(\mathfrak{h})} V
\arrow[swap]{r}{\operatorname{id} \otimes \pi} &
- \mfrac{\mathcal{U}(\mathfrak{h})}{\mathfrak{m}_\lambda}
+ \mfrac{\mathcal{U}(\mathfrak{h})}{I_\lambda}
\otimes_{\mathcal{U}(\mathfrak{h})} \mfrac{V}{W}
\end{tikzcd}
\end{center}
@@ -122,7 +120,7 @@ A particularly well behaved class of examples are the so called
\begin{example}\label{ex:laurent-polynomial-mod}
There is a natural action of \(\mathfrak{sl}_2(K)\) on the space \(K[x,
- x^{-1}]\) of Laurent polynomials given by the formulas in
+ 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}}
@@ -151,7 +149,7 @@ behavior in the following sense: if \(V\) is an irreducible weight
\(\mathfrak{g}\)-module, since \(V[\lambda] = \bigoplus_{\alpha \in Q}
V_{\lambda + \alpha}\) is stable under the action of \(\mathfrak{g}\) for all
\(\lambda \in \mathfrak{h}^*\), \(\bigoplus_{\alpha \in Q} V_{\lambda +
-\alpha}\) is either \(0\) or all \(V\). In other words, the support of an
+\alpha}\) is either \(0\) or all of \(V\). In other words, the support of an
irreducible weight module is always contained in a single \(Q\)-coset.
However, the behavior of \(K[x, x^{-1}]\) deviates from that of an arbitrary
@@ -186,8 +184,8 @@ induction functors \(\operatorname{Ind}_{\mathfrak{p}}^{\mathfrak{g}} :
\(\mathfrak{p} \subset \mathfrak{g}\) is some subalgebra. These functors have
already proved themselves a powerful tool for constructing representations in
the previous chapters. Our first observation is that if \(\mathfrak{p} \subset
-\mathfrak{g}\) contains the Borel \(\mathfrak{b}\) then \(\mathfrak{h} \subset
-\mathfrak{p}\) is a Cartan subalgebra of \(\mathfrak{p}\) and
+\mathfrak{g}\) contains the Borel subalgebra \(\mathfrak{b}\) then
+\(\mathfrak{h}\) is a Cartan subalgebra of \(\mathfrak{p}\) and
\((\operatorname{Ind}_{\mathfrak{p}}^{\mathfrak{g}} V)_\lambda =
\mathcal{U}(\mathfrak{g}) \otimes_{\mathcal{U}(\mathfrak{p})} V_\lambda\) for
all \(\lambda \in \mathfrak{h}^*\). In particular,
@@ -275,10 +273,11 @@ relationship is well understood. Namely, Fernando himself established\dots
\begin{proposition}[Fernando]
Given a parabolic subalgebra \(\mathfrak{p} \subset \mathfrak{g}\), there
exists a basis \(\Sigma\) for \(\Delta\) such that \(\Sigma \subset
- \Delta_{\mathfrak{p}'}\). 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 \(L_{\mathfrak{p}}(V) \cong
+ \Delta_{\mathfrak{p}} \subset \Delta\), where \(\Delta_{\mathfrak{p}}\)
+ denotes the set of roots of \(\mathfrak{p}\). 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 \(\mathfrak{p}' =
\mathfrak{p}^\sigma\) and \(V \cong \sigma W\) for some\footnote{Here
$\mathfrak{p}^\sigma$ denotes the image of $\mathfrak{p}$ under the
@@ -427,15 +426,16 @@ summing over \(\lambda \in K\), as in
\varphi_\lambda K[x, x^{-1}],
\]
-To a distracted spectator, \(\mathcal{M}\) may look like just another, innocent,
-\(\mathfrak{sl}_2(K)\)-module. However, the attentive reader may have already
-noticed some of the its bizarre features, most noticeable of which is the fact
-that \(\mathcal{M}\) is very big. In fact, \(\mathcal{M}\) is as big a degree
-\(1\) admissible representation gets: \(\operatorname{supp} \mathcal{M} =
-\operatorname{supp}_{\operatorname{ess}} \mathcal{M}\) is the entirety of
-\(\mathfrak{h}^*\). This should look very alien to readers familiarized with
-the theory of finite-dimensional weight modules. For this reason alone,
-\(\mathcal{M}\) deserves to be called ``a monstrous concoction''.
+To a distracted spectator, \(\mathcal{M}\) may look like just another,
+innocent, \(\mathfrak{sl}_2(K)\)-module. However, the attentive reader may have
+already noticed some of the its bizarre features, most noticeable of which is
+the fact that \(\mathcal{M}\) is very big. In fact, \(\mathcal{M}\) is as big a
+degree \(1\) admissible representation gets: \(\operatorname{supp} \mathcal{M}
+= \operatorname{supp}_{\operatorname{ess}} \mathcal{M}\) is the entirety of
+\(\mathfrak{h}^*\). This may look very alien the reader familiarized with the
+finite-dimensional setting, where the configuration of weights is very rigid.
+For this reason, \(\mathcal{M}\) deserves to be called ``a monstrous
+concoction''.
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
@@ -451,7 +451,13 @@ named \emph{coherent families}.
\item \(\dim \mathcal{M}_\lambda = d\) for \emph{all} \(\lambda \in
\mathfrak{h}^*\) -- i.e. \(\operatorname{supp}_{\operatorname{ess}}
\mathcal{M} = \mathfrak{h}^*\)
- \item For any \(u \in \mathcal{U}(\mathfrak{g})\) in the centralizer
+ \item For any \(u \in \mathcal{U}(\mathfrak{g})\) in the
+ centralizer\footnote{The notation $\mathcal{U}(\mathfrak{g})_0$ for the
+ centralizer of $\mathfrak{h}$ in $\mathcal{U}(\mathfrak{g})$ comes from
+ the fact it coincides with the weight space associated with $0 \in
+ \mathfrak{h}^*$ in the adjoint action of $\mathfrak{g}$ on
+ $\mathcal{U}(\mathfrak{g})$ -- not to be confused with the regular action
+ of $\mathfrak{g}$ on $\mathcal{U}(\mathfrak{g})$.}
\(\mathcal{U}(\mathfrak{g})_0\) of \(\mathfrak{h}\) in
\(\mathcal{U}(\mathfrak{g})\), the map
\begin{align*}
@@ -489,21 +495,22 @@ named \emph{coherent families}.
\end{example}
\begin{note}
- We should point out that coherent families have proven themselves useful for
- problems other than the classification of cuspidal \(\mathfrak{g}\)-modules.
- For instance, Nilsson's classification of rank 1 \(\mathfrak{h}\)-free
- representations of \(\mathfrak{sp}_{2 n}(K)\) is based on the notion of
- coherent families and the so called \emph{weighting functor}.
+ We would lie to stress that coherent families have proven themselves useful
+ for problems other than the classification of cuspidal
+ \(\mathfrak{g}\)-modules. For instance, Nilsson's classification of rank 1
+ \(\mathfrak{h}\)-free representations of \(\mathfrak{sp}_{2 n}(K)\) is based
+ on the notion of coherent families and the so called \emph{weighting
+ functor}.
\end{note}
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
+somehow fit \(V\) 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
+ Given an admissible \(\mathfrak{g}\)-module \(V\) of degree \(d\), a
+ \emph{coherent extension \(\mathcal{M}\) of \(V\)} is a coherent family
\(\mathcal{M}\) of degree \(d\) that contains \(V\) as a subquotient.
\end{definition}
@@ -512,7 +519,7 @@ 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
+cuspidal representations, which we now know is the key for the solution of our
classification problem. However, there are some complications to this scheme.
Leaving aside the question of existence for a second, we should point out that
@@ -543,7 +550,8 @@ to a completely reducible coherent extension of \(V\).
% including it in here
\begin{lemma}\label{thm:component-coh-family-has-finite-length}
Given a coherent family \(\mathcal{M}\) and \(\lambda \in \mathfrak{h}^*\),
- \(\mathcal{M}[\lambda]\) has finite length as a \(\mathfrak{g}\)-module.
+ \(\mathcal{M}[\lambda]\) has finite length as a
+ \(\mathcal{U}(\mathfrak{g})\)-module.
\end{lemma}
\begin{corollary}
@@ -557,12 +565,12 @@ to a completely reducible coherent extension of \(V\).
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\footnote{Notice
- that $\mathcal{M}[\lambda] = \mathcal{N}[\mu]$ for any $\mu \in \lambda + Q$.
+ that $\mathcal{M}[\lambda] = \mathcal{M}[\mu]$ for any $\mu \in \lambda + Q$.
Hence the sum $\bigoplus_{\lambda + Q \in \mfrac{\mathfrak{h}^*}{Q}}
\bigoplus_i \mfrac{\mathcal{M}_{\lambda i + 1}}{\mathcal{M}_{\lambda i}}$ is
- independent of the choice of representative for $\lambda + Q$ -- at least as
- long as we choose $\mathcal{M}_{\lambda i} = \mathcal{M}_{\mu i}$ for all
- $\mu \in \lambda + Q$ and $i$.},
+ independent of the choice of representative for $\lambda + Q$ -- provided we
+ choose $\mathcal{M}_{\mu i} = \mathcal{M}_{\lambda i}$ for all $\mu \in
+ \lambda + Q$ and $i$.},
\[
\mathcal{M}^{\operatorname{ss}}
\cong \bigoplus_{\substack{\lambda + Q \in \mfrac{\mathfrak{h}^*}{Q} \\ i}}
@@ -573,7 +581,8 @@ to a completely reducible coherent extension of \(V\).
\begin{proof}
The uniqueness of \(\mathcal{M}^{\operatorname{ss}}\) should be clear:
since \(\mathcal{M}^{\operatorname{ss}}\) is completely reducible, so is
- \(\mathcal{M}^{\operatorname{ss}}[\lambda]\). Hence
+ \(\mathcal{M}^{\operatorname{ss}}[\lambda]\). Hence by the Jordan-Hölder
+ theorem
\[
\mathcal{M}^{\operatorname{ss}}[\lambda]
\cong
@@ -652,7 +661,7 @@ promised, if \(\mathcal{M}\) is a coherent extension of \(V\) then so is
\begin{proposition}
Let \(V\) be an irreducible admissible \(\mathfrak{g}\)-module and
\(\mathcal{M}\) be a coherent extension of \(V\). Then
- \(\mathcal{M}^{\operatorname{ss}}\) is a coherent extension of \(V\) and
+ \(\mathcal{M}^{\operatorname{ss}}\) is a coherent extension of \(V\), and
\(V\) is in fact a subrepresentation of \(\mathcal{M}^{\operatorname{ss}}\).
\end{proposition}
@@ -681,24 +690,19 @@ itself and therefore\dots
\(V\) 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: which submodules of a nice
-coherent family are cuspidal representations?
+This last results provide a partial answer to the question of existence of well
+behaved coherent extensions. A complementary question now is: which submodules
+of a \emph{nice} coherent family are cuspidal representations?
\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
- centralizer\footnote{This notation comes from the fact that the centralizer of
- $\mathfrak{h}$ in $\mathcal{U}(\mathfrak{g})$ coincides the weight space
- associated with $0 \in \mathfrak{h}^*$ in the adjoint action of
- $\mathfrak{g}$ on $\mathcal{U}(\mathfrak{g})$ -- not to be confused with the
- regular action of $\mathfrak{g}$ on $\mathcal{U}(\mathfrak{g})$.} 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 \(\lambda \in
- \operatorname{supp} V\).
+ centralizer 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{corollary}[Mathieu]
@@ -818,7 +822,7 @@ deemed informative enough to be included in here, but see the proof of lemma
\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 \(\lambda\)
+ = d^2\). This goes to show that \(U\) is precisely the set of all \(\lambda\)
such that \(B_\lambda\) has maximal rank \(d^2\). We now show that \(U\) is
Zariski-open. First, notice that
\[
@@ -827,7 +831,9 @@ deemed informative enough to be included in here, but see the proof of lemma
U_W,
\]
where \(U_W = \{\lambda \in \mathcal{U}(\mathfrak{g})_0 : \operatorname{rank}
- B_\lambda\!\restriction_W = d^2 \}\).
+ B_\lambda\!\restriction_W = d^2 \}\). Here \(W\) ranges over all
+ \(d\)-dimensional subspaces of \(\mathcal{U}(\mathfrak{g})_0\) -- \(W\) is
+ not necessarily a \(\mathcal{U}(\mathfrak{g})_0\)-submodule.
Indeed, if \(\operatorname{rank} B_\lambda = d^2\) it follows from the
subjectivity of the map \(\mathcal{U}(\mathfrak{g})_0 \to
@@ -871,8 +877,8 @@ deemed informative enough to be included in here, but see the proof of lemma
the union of Zariski-open subsets and is therefore open. We are done.
\end{proof}
-The major remaining question for us to tackle is that of existence of coherent
-extensions, which will be the focus of our next section.
+The major remaining question for us to tackle is that of the existence of
+coherent extensions, which will be the focus of our next section.
\section{Localizations \& the Existence of Coherent Extensions}
@@ -892,7 +898,7 @@ injective. Since all weight spaces of \(K[x, x^{-1}]\) are 1-dimensional,
this implies the action of \(f\) is actually bijective, so we can obtain a
nonzero vector in \(K[x, x^{-1}]_{2 k} = K x^k\) for any \(k \in \mathbb{Z}\)
by translating between weight spaced using \(f\) and \(f^{-1}\) -- here
-\(f^{-1}\) denote the differential operator \((-
+\(f^{-1}\) denotes the \(K\)-linear operator \((-
\sfrac{\mathrm{d}}{\mathrm{d}x} + \sfrac{x^{-1}}{2})^{-1}\), which is the
inverse of the action of \(f\) on \(K[x, x^{-1}]\).
\begin{center}
@@ -900,23 +906,24 @@ inverse of the action of \(f\) on \(K[x, x^{-1}]\).
\cdots \arrow[bend left=60]{r}{f^{-1}}
& K x^{-2} \arrow[bend left=60]{r}{f^{-1}} \arrow[bend left=60]{l}{f}
& K x^{-1} \arrow[bend left=60]{r}{f^{-1}} \arrow[bend left=60]{l}{f}
- & K x^0 \arrow[bend left=60]{r}{f^{-1}} \arrow[bend left=60]{l}{f}
- & K x^1 \arrow[bend left=60]{r}{f^{-1}} \arrow[bend left=60]{l}{f}
+ & K \arrow[bend left=60]{r}{f^{-1}} \arrow[bend left=60]{l}{f}
+ & K x \arrow[bend left=60]{r}{f^{-1}} \arrow[bend left=60]{l}{f}
& K x^2 \arrow[bend left=60]{r}{f^{-1}} \arrow[bend left=60]{l}{f}
& \cdots \arrow[bend left=60]{l}{f}
\end{tikzcd}
\end{center}
-In our case, the action of some \(F_\alpha \in \mathfrak{g}\) with \(\alpha \in
-\Delta\) in \(V\) may not be injective. In fact, we have seen that the action
-of \(F_\alpha\) is injective for all \(\alpha \in \Delta\) if, and only if
-\(V\) is cuspidal. Nevertheless, we could intuitively \emph{make it injective}
-by formally inverting the elements \(F_\alpha \in \mathcal{U}(\mathfrak{g})\).
-This would allow us to obtain nonzero vectors in \(V_\mu\) for all \(\mu \in
-\lambda + Q\) by successively applying elements of \(\{F_\alpha^{\pm
-1}\}_{\alpha \in \Delta}\) to a nonzero weight vector \(v \in V_\lambda\).
-Moreover, if the actions of the \(F_\alpha\) were to be invertible, we would
-find that all \(V_\mu\) are \(d\)-dimensional for \(\mu \in \lambda + Q\).
+In the general case, the action of some \(F_\alpha \in \mathfrak{g}\) with
+\(\alpha \in \Delta\) in \(V\) may not be injective. In fact, we have seen that
+the action of \(F_\alpha\) is injective for all \(\alpha \in \Delta\) if, and
+only if \(V\) is cuspidal. Nevertheless, we could intuitively \emph{make it
+injective} by formally inverting the elements \(F_\alpha \in
+\mathcal{U}(\mathfrak{g})\). This would allow us to obtain nonzero vectors in
+\(V_\mu\) for all \(\mu \in \lambda + Q\) by successively applying elements of
+\(\{F_\alpha^{\pm 1}\}_{\alpha \in \Delta}\) to a nonzero weight vector \(v \in
+V_\lambda\). Moreover, if the actions of the \(F_\alpha\) were to be
+invertible, we would find that all \(V_\mu\) are \(d\)-dimensional for \(\mu
+\in \lambda + Q\).
In a commutative domain, this can be achieved by tensoring our module by the
field of fractions. However, \(\mathcal{U}(\mathfrak{g})\) is hardly ever
@@ -932,17 +939,18 @@ now describe.
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 \(r s = t_2 u_2\).
+ condition} if for each \(r \in R\) and \(s \in S\) 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 \(S\)},
+ ring homomorphism \(R \to S^{-1} R\), 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}
@@ -1007,12 +1015,11 @@ It is interesting to observe that, unlike in the case of the field of fractions
of a commutative domain, in general the localization map \(R \to S^{-1} R\) --
i.e. the map \(r \mapsto \frac{r}{1}\) -- may not be injective. For instance,
if \(S\) contains a divisor of zero \(s\), its image under the localization map
-cannot be a divisor of zero in \(S^{-1} R\) -- since it is invertible. In
-particular, if \(r \in R\) is nonzero and such that \(s r = 0\) then its image
-under the localization map has to be \(0\), given that the image of \(s r = 0\)
-is \(0\). However, the existence of divisors of zero in \(S\) turns out to be
-the only obstruction to the injectivity of the localization map, as shown
-in\dots
+is invertible and therefore cannot be a divisor of zero in \(S^{-1} R\). In
+particular, if \(r \in R\) is nonzero and such that \(s r = 0\) or \(r s = 0\)
+then its image under the localization map has to be \(0\). However, the
+existence of divisors of zero in \(S\) turns out to be the only obstruction to
+the injectivity of the localization map, as shown in\dots
\begin{lemma}
Let \(S \subset R\) be a multiplicative subset satisfying Ore's localization
@@ -1059,12 +1066,11 @@ to be included in here. See lemma 4.4 of \cite{mathieu} for a full proof. Since
\end{corollary}
From now on let \(\Sigma\) be some fixed basis for \(\Delta\) satisfying the
-hypothesis of lemma~\ref{thm:nice-basis-for-inversion}. As promise, we now show
-that \(\Sigma^{-1} V\) contains \(V\) and that its support is an entire
-\(Q\)-coset.
+hypothesis of lemma~\ref{thm:nice-basis-for-inversion}. We now show that
+\(\Sigma^{-1} V\) is a weight module whose support is an entire \(Q\)-coset.
\begin{proposition}\label{thm:irr-admissible-is-contained-in-nice-mod}
- The the restriction of the localization \(\Sigma^{-1} V\) is an admissible
+ 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 \(\lambda \in \operatorname{supp} \Sigma^{-1} V\).
@@ -1072,10 +1078,10 @@ that \(\Sigma^{-1} V\) contains \(V\) and that its support is an entire
\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
+ \(F_\beta^{-1}\) map the weight space \(\Sigma^{-1} V_\lambda\) to
+ \(\Sigma^{-1} V_{\lambda - \beta}\) 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
@@ -1126,14 +1132,15 @@ that \(\Sigma^{-1} V\) contains \(V\) and that its support is an entire
We now have a good candidate for a coherent extension of \(V\), but
\(\Sigma^{-1} V\) is still not a coherent extension since its support is
-contained in a single coset. In particular, \(\operatorname{supp} \Sigma^{-1} V
-\ne \mathfrak{h}^*\) and \(\Sigma^{-1} V\) is not a coherent family. To obtain
-a coherent family we thus need somehow extend \(\Sigma^{-1} V\). To that end,
-we'll attempt to replicate the construction of the coherent extension of the
-\(\mathfrak{sl}_2(K)\)-module \(K[x, x^{-1}]\). Specifically, the idea is that
-if twist \(\Sigma^{-1} V\) by an automorphism which shifts its support by some
-\(\lambda \in \mathfrak{h}^*\), we can construct a coherent family by summing
-this modules over \(\lambda\) as in example~\ref{ex:sl-laurent-family}.
+contained in a single \(Q\)-coset. In particular, \(\operatorname{supp}
+\Sigma^{-1} V \ne \mathfrak{h}^*\) and \(\Sigma^{-1} V\) is not a coherent
+family. To obtain a coherent family we thus need somehow extend \(\Sigma^{-1}
+V\). To that end, we'll attempt to replicate the construction of the coherent
+extension of the \(\mathfrak{sl}_2(K)\)-module \(K[x, x^{-1}]\). Specifically,
+the idea is that if twist \(\Sigma^{-1} V\) by an automorphism which shifts its
+support by some \(\lambda \in \mathfrak{h}^*\), we can construct a coherent
+family by summing this modules over \(\lambda\) as in
+example~\ref{ex:sl-laurent-family}.
For \(K[x, x^{-1}]\) this was achieved by twisting the
\(\operatorname{Diff}(K[x, x^{-1}])\)-module \(K[x, x^{-1}]\) by the
@@ -1145,9 +1152,9 @@ this approach is inflexible since not every representation of
\(\mathfrak{sl}_2(K)\) factors through \(\operatorname{Diff}(K[x, x^{-1}])\).
Nevertheless, we could just as well twist \(K[x, x^{-1}]\) by automorphisms of
\(\mathcal{U}(\mathfrak{sl}_2(K))_f\) directly -- where
-\(\mathcal{U}(\mathfrak{sl}_2(K))_f\) denotes the localization of
-\(\mathcal{U}(\mathfrak{sl}_2(K))\) by the multiplicative subset generated by
-\(f\).
+\(\mathcal{U}(\mathfrak{sl}_2(K))_f = (f)^{-1} \mathcal{U}(\mathfrak{g})\) is
+the localization of \(\mathcal{U}(\mathfrak{sl}_2(K))\) by the multiplicative
+subset generated by \(f\).
In general, we may twist the \(\Sigma^{-1} \mathcal{U}(\mathfrak{g})\)-module
\(\Sigma^{-1} V\) by automorphisms of \(\Sigma^{-1}
@@ -1310,9 +1317,9 @@ It should now be obvious\dots
It is clear \(V\) lies in \(\Sigma^{-1} V = \theta_0 \Sigma^{-1} V\) 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 \(\mu \in \lambda + Q\), \(\lambda \in
- \Lambda\). Furthermore, given \(u \in \mathcal{U}(\mathfrak{g})_0\) and
- \(\mu \in \lambda + Q\),
+ V_{\mu - \lambda} = d\) for all \(\mu \in \lambda + Q\) -- \(\lambda\)
+ standing for some fixed representative of its \(Q\)-coset. Furthermore, given
+ \(u \in \mathcal{U}(\mathfrak{g})_0\) and \(\mu \in \lambda + Q\),
\[
\operatorname{Tr}(u\!\restriction_{\mathcal{M}_\mu})
= \operatorname{Tr}
@@ -1382,7 +1389,7 @@ Lo and behold\dots
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
+ are polynomial in \(\lambda \in \mathfrak{h}^*\), it follows that these maps
coincide for all \(u\).
In conclusion, \(\mathcal{N} \cong \operatorname{Ext}(V)\) and
@@ -1407,16 +1414,16 @@ the subject of sections 7, 8 and 9 of \cite{mathieu}. In addition, in sections
coherent families. We unfortunately do not have the necessary space to discuss
these results in detail, but we will now provide a brief overview.
-First and foremost, the problem of classifying the coherent
-\(\mathfrak{g}\)-family can be reduced to that of classifying only the coherent
-\(\mathfrak{sl}_n(K)\)-families and coherent \(\mathfrak{sp}_{2
-n}(K)\)-families. This is because of the following results.
+First and foremost, the problem of classifying \(\mathfrak{g}\)-family can be
+reduced to that of classifying only \(\mathfrak{sl}_n(K)\)-families and
+coherent \(\mathfrak{sp}_{2 n}(K)\)-families. This is because of the following
+results.
\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 \(V\) can be decomposed as
+ and \(\mathfrak{s}_1, \ldots, \mathfrak{s}_n\) are its simple component, then
+ any irreducible weight \(\mathfrak{g}\)-module \(V\) can be decomposed as
\[
V \cong Z \boxtimes V_1 \boxtimes \cdots \boxtimes V_n
\]
@@ -1431,14 +1438,15 @@ n}(K)\)-families. This is because of the following results.
\mathfrak{sp}_{2 n}(K)\).
\end{proposition}
-We've previously seen that the representation of Abelian Lie algebras are well
-understood, so to classify the irreducible representations of an arbitrary
-reductive algebra it suffices to classify those of its simple components. To
-classify these representations we can apply Fernando's results and reduce the
-problem to constructing the cuspidal representation of the simple Lie algebras.
-But by proposition~\ref{thm:only-sl-n-sp-have-cuspidal} only
-\(\mathfrak{sl}_n(K)\) and \(\mathfrak{sp}_{2 n}(K)\) admit cuspidal
-representation, so it suffices to consider these two cases.
+We've previously seen that the representations of Abelian Lie algebras,
+particularly the 1-dimensional ones, are well understood. Hence to classify the
+irreducible representations of an arbitrary reductive algebra it suffices to
+classify those of its simple components. To classify these representations we
+can apply Fernando's results and reduce the problem to constructing the
+cuspidal representation of the simple Lie algebras. But by
+proposition~\ref{thm:only-sl-n-sp-have-cuspidal} only \(\mathfrak{sl}_n(K)\)
+and \(\mathfrak{sp}_{2 n}(K)\) admit cuspidal representation, so it suffices to
+consider these two cases.
Finally, we apply Mathieu's results to further reduce the problem to that of
classifying the simple completely reducible coherent families of
@@ -1459,7 +1467,7 @@ 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 its left is to wonder at the beauty
+Alas, our journey has come to an end. All it's left is to wonder at the beauty
of Lie algebras and their representations.
\label{end-47}