Exercise 10, § 13

The notations are those of no. 4.IV, type $D_l(l\geq 2)$.

a) Let $x$ and $y$ be the elements of $\bigwedge^lV$ defined by

$x=e_1\wedge \cdots \wedge e_{l-1}\wedge e_l$ and $y=e_1\wedge \cdots \wedge e_{l-1}\wedge e_{-l}$.

The element $x$ is primitive of weight $2\varpi_l$ and $y$ is primitive of weight $2\varpi_{l-1}$. The submodule X (resp. Y) of $\bigwedge^lV$ generated by $x$ (resp. $y$) is isomorphic to $E(2\varpi_l)$ (resp. $E(2\varpi_{l-1})$). Show, by calculating dimensions, that $\bigwedge^lV =$ $X\oplus Y$; in particular, $\bigwedge^lV$ is the sum of two non-isomorphic simple modules. b) Let $e=e_1\wedge \cdots \wedge e_l\wedge e_{-1}\wedge \cdots \wedge e_{-l}\in \bigwedge^{2l}V$, and $\Psi_l$ the extension of $\Psi$ to $\bigwedge^lV$. Let $z\in \bigwedge^lV$. Prove the equivalences:

$z\in X\Leftarrow \Rightarrow z\wedge t=\Psi_l(z, t)e$ for all $t\in \bigwedge^lV$

$z\in Y\Leftarrow \Rightarrow z\wedge t=-\Psi_l(z, t)e$ for all $t\in \bigwedge^lV$.

(If $X'$ and $Y'$ denote the subspaces defined by the right-hand sides, prove first that $X'$ and $Y'$ are stable under $\mathfrak{g}$ and contain $x$ and $y$, respectively.) c) Assume that $z$ is pure (Algebra, Chap. III, §11, no. 13), and denote by $M_z$ the $l$-dimensional subspace of V associated to it. Show that $M_z$ is totally isotropic if and only if $z$ belongs to X or to Y. (Use the fact that, when $M_z$ is totally isotropic, there exists an orthogonal transformation that transforms it into $M_x$; when $M_z$ is not totally isotropic, construct an $l$-vector $t$ such that $z\wedge t= 0, \Psi_l(z, t) = 1$, and apply b) above.) When $z\in X$ (resp. $z\in Y$), the dimension of $M_x/(M_x\cap M_z)$ is an even (resp. odd) integer, cf. Algebra, Chap. IX, §6, Exerc. $18d)$.

d) Let $s$ be a direct (resp. inverse) similarity of V. Show that $\bigwedge^ls$ leaves X and Y stable (resp. interchanges X and Y).