Exercise 8, § 7

Let $\mathfrak{g}$ be a semi-simple Lie algebra; exceptionally, we do not assume in this exercise that $\mathfrak{g}$ is split. Let $\overline{k}$ be an algebraic closure of $k$ and let

$\pi :$ Gal($\overline{k}/k$)$\rightarrow$ Aut( $\overline{R},\overline{B}$)

be the homomorphism defined in §5, Exerc. 8. Let Gal($\overline{k}/k$) operate, via $\pi$, on the set $\overline{P}_{++}$ of dominant weights of $\overline{R}$ relative to $\overline{B}$; let $\Omega$ be a system of representatives of the elements of the quotient $\overline{P}_{++}/$Gal($\overline{k}/k$).

a) Put $\overline{\mathfrak{g}}= \overline{k}\otimes_k\mathfrak{g}$. If I is a finite subset of $\overline{P}_{++}$ stable under Gal($\overline{k}/k$), the two-sided ideal $\overline{\mathfrak{m}}_I$ of $U(\overline{\mathfrak{g}})$ associated to I (cf. Exerc. 7) is of the form $\overline{k}\otimes_k\mathfrak{m}_I$, where $\mathfrak{m}_I$ is a two-sided ideal of $U(\mathfrak{g})$. Show that every two-sided ideal of $U(\mathfrak{g})$ of finite codimension is obtained in this way exactly once.

b) Let $\omega \in \Omega , I(\omega )$ its orbit under Gal($\overline{k}/k$), and $G_{\omega}$ the stabilizer of $\omega$; let $k_{\omega}$ be the sub-extension of $\overline{k}$ corresponding to $G_{\omega}$ by Galois theory. Show that $U(\mathfrak{g})/\mathfrak{m}_{I(\omega)}$ is a simple algebra whose centre is isomorphic to $k_{\omega}$. Every two-sided ideal $\mathfrak{m}$ of $U(\mathfrak{g})$ such that $U(\mathfrak{g})/\mathfrak{m}$ is a finite dimensional simple algebra is equal to exactly one of the $\mathfrak{m}_{I(\omega)}$.

c) The group Gal($\overline{k}/k$) operates, via $\pi$, on the ring $R(\overline{\mathfrak{g}})$; denote by $R(\overline{\mathfrak{g}})^{inv}$ the subring of $R(\overline{\mathfrak{g}})$ consisting of the elements invariant under Gal($\overline{k}/k$). Show that the map $[E]\rightarrow [\overline{k}\otimes_kE]$ extends to an injective homomorphism from $R(\mathfrak{g})$ to $R(\overline{\mathfrak{g}})^{inv}$ whose cokernel is a torsion group; this is an isomorphism if and only if, for all $\omega \in \Omega , U(\mathfrak{g})/\mathfrak{m}_{I(\omega)}$ is an algebra of matrices over $k_{\omega}$; show that this is the case when $\mathfrak{g}$ has a Borel subalgebra. $^{14}$

$^{14}$For more details on this exercise, see: J. TITS, Représentations linéaires irréductibles d’un groupe réductif sur un corps quelconque, J. für die reine und angewandte Math. Vol. CCXLVII (1971), pp. 196-220.