diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -136,12 +136,12 @@ A particularly well behaved class of examples are the so called
Hence \(W = K[x, x^{-1}]\) and \(K[x, x^{-1}]\) is an irreducible
representation.
\begin{align}\label{eq:laurent-polynomials-cusp-mod}
+ e \cdot p
+ & = \left( x^2 \frac{\mathrm{d}}{\mathrm{d}x} + \frac{x}{2} \right) p &
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
+ & = 2 x \frac{\mathrm{d}}{\mathrm{d}x} p
\end{align}
\end{example}
@@ -407,16 +407,16 @@ x^{-1}])\) are the ones from the previous diagram.
Explicitly, we find that the action of \(\mathfrak{sl}_2(K)\) in
\(\varphi_\lambda K[x, x^{-1}]\) is given by
\begin{align*}
+ p & \overset{e}{\mapsto}
+ \left(
+ x^2 \frac{\mathrm{d}}{\mathrm{d}x} + \frac{1 + \lambda}{2} x
+ \right) p &
p & \overset{f}{\mapsto}
\left(
- \frac{\mathrm{d}}{\mathrm{d}x} + \frac{1 - \lambda}{2} x^{-1}
\right) p &
p & \overset{h}{\mapsto}
- \left( 2 x \frac{\mathrm{d}}{\mathrm{d}x} + \lambda \right) p &
- p & \overset{e}{\mapsto}
- \left(
- x^2 \frac{\mathrm{d}}{\mathrm{d}x} + \frac{1 + \lambda}{2} x
- \right) p,
+ \left( 2 x \frac{\mathrm{d}}{\mathrm{d}x} + \lambda \right) p,
\end{align*}
so we can see \((\varphi_\lambda K[x, x^{-1}])_{2 k + \frac{\lambda}{2}} = K
x^k\) for all \(k \in \mathbb{Z}\) and \((\varphi_\lambda K[x, x^{-1}])_\mu =
@@ -483,16 +483,21 @@ named \emph{coherent families}.
\begin{example}
Given \(\lambda \in K\), \(\mathcal{M}(\lambda) = \bigoplus_{\mu \in K} K
- m_\mu\) with
+ x^\mu\) with
\begin{align*}
- m_\mu & \overset{f}{\mapsto} (\lambda - \mu) m_{\mu - 1} &
- m_\mu & \overset{h}{\mapsto} 2\mu m_\mu &
- m_\mu & \overset{e}{\mapsto} (\lambda + \mu) m_{\mu + 1},
+ p & \overset{e}{\mapsto}
+ \left(x^2 \frac{\mathrm{d}}{\mathrm{d}x} + \lambda x\right) p &
+ p & \overset{f}{\mapsto}
+ \left(-\frac{\mathrm{d}}{\mathrm{d}x} + \lambda x^{-1}\right) p &
+ p & \overset{h}{\mapsto} 2 x \frac{\mathrm{d}}{\mathrm{d}x} p,
\end{align*}
- is a degree \(1\) coherent \(\mathfrak{sl}_2(K)\)-family. It is easy to check
- \(\mathcal{M}\) from example~\ref{ex:sl-laurent-family} is isomorphic to
- \(\mathcal{M}(\sfrac{1}{2})\). In fact by rewriting the symbol \(m_k\) as
- \(x^k\) we can see \((\mathcal{M}(\sfrac{1}{2}))[0] \cong K[x, x^{-1}]\).
+ is a degree \(1\) coherent \(\mathfrak{sl}_2(K)\)-family -- where \(x^{\pm
+ 1}, \sfrac{\mathrm{d}}{\mathrm{d}x} : \mathcal{M}(\lambda) \to
+ \mathcal{M}(\lambda)\) are given by \(x^{\pm 1} x^\mu = x^{\mu \pm 1}\) and
+ \(\sfrac{\mathrm{d}}{\mathrm{d}x} x^\mu = \mu x^{\mu - 1}\). It is easy to
+ check \(\mathcal{M}\) from example~\ref{ex:sl-laurent-family} is isomorphic
+ to \(\mathcal{M}(\sfrac{1}{2})\) and \((\mathcal{M}(\sfrac{1}{2}))[0] \cong
+ K[x, x^{-1}]\).
\end{example}
Our hope is that given an irreducible cuspidal representation \(V\), we can