Exercise 15, § 13

Let V be a vector space of finite dimension $\geq 2,\mathfrak{g}$ a Lie subalgebra of End(V) and $\theta$ an element of $\mathfrak{g}$. We make the following assumptions:

(i) V is a semi-simple $\mathfrak{g}$-module; (ii) $\theta$ is of rank 1 (i.e. dim Im($\theta$ ) $= 1$);

(iii) the line Im($\theta$ ) generates the $U(\mathfrak{g}$)-module V.

a) Show that these assumptions are satisfied (for a suitable choice of $\theta$ ) when $\mathfrak{g}=\mathfrak{s}\mathfrak{l}$(V), when $\mathfrak{g}=\mathfrak{g}\mathfrak{l}$(V), or when there exists a non-degenerate alternating bilinear form $\Psi$ on V such that $\mathfrak{g}=\mathfrak{s}\mathfrak{p}(\Psi )$ or $\mathfrak{g}=k.1\oplus \mathfrak{s}\mathfrak{p}(\Psi )$. In each of these cases, $\theta$ can be taken to be a nilpotent element; in the second case (and only in that case) $\theta$ can be taken to be a semi-simple element.

b) We shall prove that the four cases above are the only ones possible. We are reduced immediately to the case in which $k$ is algebraically closed. Show that V is then a simple $\mathfrak{g}$-module, and that $\mathfrak{g}=\mathfrak{c}\oplus \mathfrak{s}$, where $\mathfrak{s}$ is semi-simple, and $\mathfrak{c}= 0$ or $\mathfrak{c}=k.1$; show that V is not isomorphic to a tensor product of $\mathfrak{g}$-modules of dimension $\geq 2$; deduce that $\mathfrak{s}$ is simple.

c) Choose a Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{s}$, and a basis B of $R(\mathfrak{s},\mathfrak{h})$. Let $\lambda$ be the highest weight (with respect to B) of the $\mathfrak{s}$-module V, and let $e$ be a non-zero element of V of weight $\lambda$. The highest weight of the dual module $V^*$ is $\lambda^*=-w_0\lambda ($§7, no. 5); let $e^*$ be a non-zero element of $V^*$ of weight $\lambda^*$. Identify $V\otimes V^*$ with End(V) as usual. Show that there exist $x\in V, y\in V^*$ such that $x\otimes y\in \mathfrak{g}$ and $\langle x, e^*\rangle \not= 0,\langle e, y\rangle \not= 0$ (take the conjugate of $\theta$ by $e^n$, where $n$ is a suitable nilpotent element of $\mathfrak{g}$). Use the fact that $\mathfrak{g}$ is an $\mathfrak{h}$-submodule of $V\otimes V^*$ to conclude that $\mathfrak{g}$ contains $e\otimes e^*$. Deduce that $\lambda +\lambda^*= \widetilde{\alpha}$, where $\widetilde{\alpha}$ is the highest root of $\mathfrak{s}$.

d) Show that $\mathfrak{s}$ is not of type $B_l(l\geq 2), D_l(l\geq 4), E_6,E_7,E_8,F_4,G_2$ (by Chap. VI, Tables$, \widetilde{\alpha}$ would be a fundamental weight, and hence could not be of the form $\lambda +\lambda^*$ above). Deduce that $\mathfrak{s}$ is either of type $A_l$ or of type $C_l$, and that in the first case $\lambda =\varpi_1$ or $\varpi_1=\varpi_1^*$, and in the second case $\widetilde{\alpha}= 2\varpi_1=\varpi_1+\varpi^*_1$; since $\mathfrak{c}= 0$ or $k.1$, this indeed gives the four possibilities in $a)^{21}$.