- Commit
- 21223791c87bbd86c80de8d75793a776be6d7b03
- Parent
- 6456f3ea14bfc85db5ebf3fb6fb285b30f3e0051
- Author
- Pablo <pablo-escobar@riseup.net>
- Date
Changed the notation for the twisting isomorphisms
Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules
Changed the notation for the twisting isomorphisms
1 file changed, 10 insertions, 10 deletions
Status | File Name | N° Changes | Insertions | Deletions |
Modified | sections/mathieu.tex | 20 | 10 | 10 |
diff --git a/sections/mathieu.tex b/sections/mathieu.tex @@ -343,12 +343,12 @@ % TODO: Remark that any module over the localization is a g-module if we % restrict it via the localization map, wich is injective in this case \begin{proposition}\label{thm:nice-automorphisms-exist} - There is a family of automorphisms \(\{F_{\Sigma}^\lambda : + There is a family of automorphisms \(\{\theta_\lambda : \mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in \Sigma)} \to \mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in \Sigma)}\}_{\lambda \in \mathfrak{h}^*}\) such that \begin{enumerate} - \item \(F_{\Sigma}^{k_1 \alpha_1 + \cdots k_n \alpha^n}(u) = + \item \(\theta_{k_1 \alpha_1 + \cdots k_n \alpha^n}(u) = F_{\alpha_1}^{k_1} \cdots F_{\alpha_n}^{k_n} u F_{\alpha_1}^{- k_n} \cdots F_{\alpha_n}^{- k_1}\) for all \(u \in \mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in \Sigma)}\) and \(k_1, @@ -359,16 +359,16 @@ \begin{align*} \mathfrak{h}^* & \to \mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in \Sigma)} \\ - \lambda & \mapsto F_\Sigma^\lambda(u) + \lambda & \mapsto \theta_\lambda(u) \end{align*} is polynomial. \item If \(M\) is the \(\mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in \Sigma)}\)-module \(\mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in \Sigma)} \otimes V\), \(\lambda, \mu \in \mathfrak{h}^*\) and - \(F_\Sigma^\lambda M\) is the \(\mathcal{U}(\mathfrak{g})_{(F_\alpha : + \(\theta_\lambda M\) is the \(\mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in \Sigma)}\)-module \(M\) twisted by the automorphism - \(F_\Sigma^\lambda\) then \(M_\mu = (F_\Sigma^\lambda M)_{\mu + + \(\theta_\lambda\) then \(M_\mu = (\theta_\lambda M)_{\mu + \lambda}\). \end{enumerate} \end{proposition} @@ -388,18 +388,18 @@ proposition~\ref{thm:irr-admissible-is-contained-in-nice-mod} and take \[ \mathcal{M} - = \bigoplus_{\lambda \in \Lambda} F_\Sigma^\lambda M + = \bigoplus_{\lambda \in \Lambda} \theta_\lambda M \] - On the one hand, \(V\) lies in \(M = F_\Sigma^0 M\) -- recall that - \(F_\Sigma^0\) is just the identity operator -- and therefore \(V \subset + On the one hand, \(V\) lies in \(M = \theta_0 M\) -- recall that + \(\theta_0\) is just the identity operator -- and therefore \(V \subset \mathcal{M}\). On the other hand, \(\dim \mathcal{M}_\mu = \dim - F_\Sigma^\lambda M_\mu = \dim M_{\mu - \lambda} = d\) for all \(\mu \in + \theta_\lambda M_\mu = \dim M_{\mu - \lambda} = d\) for all \(\mu \in \lambda + Q\), \(\lambda \in \Lambda\). Furtheremore, given \(u \in C_{\mathcal{U}(\mathfrak{g})}(\mathfrak{h})\) and \(\mu \in \lambda + Q\), \[ \operatorname{Tr}(u\!\restriction_{\mathcal{M}_\mu}) - = \operatorname{Tr}(F_\Sigma^\lambda(u)\!\restriction_{M_{\mu - \lambda}}) + = \operatorname{Tr}(\theta_\lambda(u)\!\restriction_{M_{\mu - \lambda}}) \] is polynomial in \(\mu\) because of the second item of proposition~\ref{thm:nice-automorphisms-exist}.