- 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
Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules
Reordered some equations
Also changed the notation for an example of coherent sl2-family
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