Lemma 2

For $1\leq r\leq\ell$, let $F_r$ be the subspace of $\bigwedge^r V$ generated by the isotropic $r$-vectors. Then

We show first of all how Lemma 2 implies our assertion. Since $F_r\subset E_r$ and $E_r\cap X_-\left(\bigwedge^{r-2}V\right)=\{0\}$, the lemma implies that $F_r=E_r$. On the other hand, let $s\in\mathbf{Sp}(\Psi)$; the automorphism $a\mapsto sas^{-1}$ of $\operatorname{End}(V)$ preserves $\mathfrak{g}$ and induces on it an element of $\operatorname{Aut}_0(\mathfrak{g})$ (cf. (§7, no. 1, Prop. 2), and hence transforms every irreducible representation of $\mathfrak{g}$ into an equivalent representation (§7, no. 1, Prop. 2). Since $e_1\wedge\cdots\wedge e_r$ belongs to an irreducible component of $\bigwedge^r\sigma$, and since $E_r=F_r$ is generated by the transforms of $e_1\wedge\cdots\wedge e_r$ by $\mathbf{Sp}(\Psi)$, it follows that the representation of $\mathfrak{g}$ on $E_r$ is isotypical. But the multiplicity of its highest weight $\overline{\varpi}_r$ is $1$, so it is irreducible.

It remains to prove the lemma. It is clear for $r=1$. We argue by induction, and assume that $r\geq 2$. By the induction hypothesis, we are reduced to proving that

or that, if $y$ is a decomposable $(r-1)$-vector and $x\in V$, then

Let $(f_i)_{1\leq\pm i\leq\ell}$ be a Witt basis of $V$ such that $y=f_1\wedge\cdots\wedge f_{r-1}$. It suffices to carry out the proof when $x=f_i$. If $i\notin[1-r,-1]$, the $r$-vector $f_1\wedge\cdots\wedge f_{r-1}\wedge f_i$ is isotropic. Otherwise, we can assume, renumbering the $f_i$ if necessary, that $i=1-r$. Then $\Gamma=\displaystyle\sum_{j=1}^{\ell}f_j\wedge f_{-j}$, so

But

and we verify immediately that the $r$-vectors

$f_1\wedge \cdots \wedge f_{r-2}\wedge f_j\wedge f_{1-r}$ and $f_1\wedge \cdots \wedge f_{r-1}\wedge f_{-j}$

are isotropic for $r\leq j\leq l$. Consequently, $z\in F_r+\Gamma \wedge \bigwedge^{r-2}V$, which completes the proof.

(V) We have $w_0=-1$, so every finite dimensional simple representation of $\mathfrak{g}$ is orthogonal or symplectic. By Chap. VI, §4, no. 6.VI, the sum of the coordinates of $\varpi_r$ with respect to $(\alpha_1, . . . , \alpha_l)$ is

2

so $\sigma_r$ is orthogonal for $r$ even and symplectic for $r$ odd.

Since $e_1\wedge \cdots \wedge e_r$ and $e_{-1}\wedge \cdots \wedge e_{-r}$ belong to $E_r$ and since

we see that the restriction of $\Psi_{(r)}$ to $E_r$ is non-zero: this is, up to a constant factor, the bilinear form; it is symmetric if $r$ is even, alternating if $r$ is odd, and invariant under $\sigma_r$.

(VI) For all $x\in \mathfrak{g}$, the characteristic polynomial of $\sigma (x)$ takes the form

where $f_1, . . . , f_{2l}$ are invariant polynomial functions on $\mathfrak{g}$.

If $x=\xi_1H_1+\cdots +\xi_lH_l\in \mathfrak{h}$, the $f_i(x)$ are, up to sign, the elementary symmetric functions of $\xi_1, . . . , \xi_l,-\xi_1, . . . ,-\xi_l$; these symmetric functions are zero in odd degrees, and

As in no. 2.VI, it follows that $f_1=f_3=f_5=\cdots = 0$, and that

is an algebraically free family generating the algebra of invariant polynomial functions on $\mathfrak{g}$.

(VII) Since the only automorphism of the Dynkin graph is the identity, we have Aut($\mathfrak{g}$) $=$ Aut$_0(\mathfrak{g})$.

Let $\Sigma$ be the group of similarities of V relative to $\Psi ($Algebra, Chap. IX, §6, end of no. 5). One proves as in no. 2.VII that the automorphisms of $\mathfrak{g}$ are the maps $x \rightarrow sxs^{-1}$ where $s\in \Sigma$, so that Aut($\mathfrak{g}$) $=$ Aut$_0(\mathfrak{g})$ can be identified with $\Sigma /k^*$.

For all $s\in \Sigma$, let $\mu(s)$ be the multiplier of $s$. The map $s \rightarrow \mu(s)$ mod $k^{*2}$ from $\Sigma$ to $k^*/k^{*2}$ is a homomorphism whose kernel contains $k^*.1$, and hence gives a homomorphism $\lambda$ from $\Sigma /k^*$ to $k^*/k^{*2}$. We have $\mathbf{S}\mathbf{p}(\Psi )\cap k^*=\{1,-1\}$. Consider the sequence of homomorphisms

1 $\longrightarrow \mathbf{S}\mathbf{p}(\Psi )/\{1,-1\}\longrightarrow^{\iota}\Sigma /k^*\longrightarrow^{\lambda}k^*/k^{*2}\longrightarrow 1$. (10) The map $\iota$ is injective, and Im($\iota$ )$\subset$ Ker$\lambda$ since the multiplier of an element of $\mathbf{S}\mathbf{p}(\Psi )$ is 1. If the multiplier of $s\in \Sigma$ is an element of $k^{*2}$, there exists $\nu \in k^*$ such that $\nu s\in \mathbf{S}\mathbf{p}(\Psi )$; thus, Im($\iota$ ) $=$ Ker($\lambda$ ). In summary, the sequence (10) is exact. We identify $\mathbf{S}\mathbf{p}(\Psi )/\{1,-1\}$ with a subgroup of $\Sigma /k^*$. Since $k^*/k^{*2}$ is commutative, $\mathbf{S}\mathbf{p}(\Psi )/\{1,-1\}$ contains the derived group of $\Sigma /k^*$. Thus, Aut$_e(\mathfrak{g})$ is contained in $\mathbf{S}\mathbf{p}(\Psi )/\{1,-1\}($§11, no. 2, Prop. 3). In fact, it is equal to it, and Aut($\mathfrak{g}$)$/$Aut$_e(\mathfrak{g})$ is identified with $k^*/k^{*2}$ (Exerc. 9).

(VIII) The canonical bilinear form $\Phi_R$ on $\mathfrak{h}^*$ is given by

(Chap. VI, §4, no. 6.V). Thus, the inverse form of $\Phi_R$, that is, the restriction to $\mathfrak{h}$ of the Killing form, is

(IX) Recall the $X_{\alpha}$ defined by formulas $(6) (\alpha \in R)$. It is easily verified that $[X_{\alpha}, X_{-\alpha}] =-H_{\alpha}$ for $\alpha \in R$. On the other hand, the map $\theta :a \rightarrow -^ta$ is an automorphism of $\mathfrak{g}$ and $\theta (X_{\alpha}) =X_{-\alpha}$ for all $\alpha \in R$. Consequently, $(X_{\alpha})_{\alpha\in R}$ is a Chevalley system in $(\mathfrak{g},\mathfrak{h})$.

Assume that $k=\mathbf{Q}$. The Cartan subalgebra $\mathfrak{h}$ has two permissible lattices $Q(R^{\vee}) =\sum_{i=1}^l\mathbf{Z}.H_i$ and $P(R^{\vee}) = Q(R^{\vee}) +\frac{1}{2}\mathbf{Z}.\sum_{i=1}^lH_i$ (Chap. VI, §4,

no. 5.VIII). We see that $Q(R^{\vee})$ is the set of matrices with integer entries belonging to $\mathfrak{h}$. It follows that the Chevalley order $Q(R^{\vee}) +\sum_{\alpha\in R}\mathbf{Z}.X_{\alpha}$ is the set $\mathfrak{s}\mathfrak{p}(2l,\mathbf{Z})$ of matrices in $\mathfrak{g}$ with integer entries.

Consider the reductive Lie algebra $\mathfrak{s}\mathfrak{p}(\Psi ) +\mathbf{Q}.1$. It is easy to see that the set of its elements with integer entries is a Chevalley order, whose projection onto $\mathfrak{s}\mathfrak{p}(\Psi )$ parallel to $\mathbf{Q}.1$ is the Chevalley order $P(R^{\vee}) +\sum\mathbf{Z}.X_{\alpha}$.

Finally, $X_{\alpha}^2= 0$ for all $\alpha \in R$. It follows that the lattice $\mathscr{V}$ in V generated by the $e_i$ is admissible for the Chevalley order $\mathfrak{s}\mathfrak{p}(2l,\mathbf{Z})$. The same holds for the lattice $E_r\cap \bigwedge^r\mathscr{V}$ in $E_r$.

Finally, $E_r$ has an admissible lattice for the Chevalley order

only if $r$ is even; then $E_r\cap \bigwedge^r\mathscr{V}$ is such a lattice.