Exercise 27, § 7

Let $\Omega = P/Q$. If $\omega \in \Omega$, and if E is a $\mathfrak{g}$-module, denote by $E_{\omega}$ the direct sum of the $E^{\lambda}$, for $\lambda \in \omega$. We have $E =\bigoplus_{\omega\in\Omega}E_{\omega}$.

a) Show that $E_{\omega}$ is a $\mathfrak{g}$-submodule of E. We have $(E^*)_{\omega}= (E_{-\omega})^*$ and

if F is another $\mathfrak{g}$-module.

b) Let $\chi \in$ Hom($\Omega , k^*$) $=$ Ker(T$_P\rightarrow T_Q$). Identify $\chi$ with an element of the kernel of $f: G\rightarrow$ Aut$_e(\mathfrak{g})$, cf. Exerc. $26e)$. Show that the operation of $\chi$ on $E_{\omega}$ is the homothety of ratio $\chi (\omega )$.

c) What are the $E_{\omega}$ when $\mathfrak{g}=\mathfrak{s}\mathfrak{l}(2, k)$ ?