lie-algebras-and-their-representations

Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules

Commit
a4e3dcf245e59a78908cf7e4a88ad05b7cfd22db
Parent
fc3472fbf63d8b5cb6fae361240af3b5b77d3ce5
Author
Pablo <pablo-escobar@riseup.net>
Date

Reordered some equations

Also changed the notation for an example of coherent sl2-family

Diffstat

1 file changed, 21 insertions, 16 deletions

Status File Name N° Changes Insertions Deletions
Modified sections/mathieu.tex 37 21 16
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