(i) All the weights of V belong to P.

(ii) $V =\bigoplus_{\mu\in P}V^\mu$.

(iii) For all $\mu\in \mathfrak{h}^*,V^\mu$ is the set of $x\in V$ such that $h.x=\mu(h)x$ for all $h\in \mathfrak{h}$.

For all $\alpha \in B$, there exists a homomorphism from $\mathfrak{s}\mathfrak{l}(2, k)$ to $\mathfrak{g}$ that takes $H$ to $H_{\alpha}$. Thus, by §1, no. 2, Cor. of Prop. $2, (H_{\alpha})_V$ is diagonalizable and its eigenvalues are integers. Hence, the set of $(H_{\alpha})_V$, for $\alpha \in B$, is diagonalizable (Algebra, Chap. VII, §5, no. 6, Prop. 13). Consequently, for all $h\in \mathfrak{h},h_V$ is diagonalizable. By Chap. VII, §1, no. 3, Prop. $9, V =\bigoplus_{\mu\in\mathfrak{h}^*}V^\mu$. On the other hand, if $V^\mu\not= 0$, the preceding shows that $\mu(H_{\alpha})\in \mathbf{Z}$ for all $\alpha \in B$, so $\mu\in P$. This proves (i) and (ii). We see in the same way that $\mathfrak{h}_V$ is diagonalizable, hence (iii).

(i) If $x, y\in \mathfrak{h}_{\mathbf{Q}}$, then $\Phi (x, y)\in \mathbf{Q}$ and $\Phi (x, x)\in \mathbf{Q}_+$.

(ii) If $\rho$ is injective, the restriction of $\Phi$ to $\mathfrak{h}$ is non-degenerate.

Assertion (i) follows from Prop. 1 since the elements of P have rational values on $\mathfrak{h}_{\mathbf{Q}}$. If $\rho$ is injective, $\Phi$ is non-degenerate (Chap. I, §6, no. 1, Prop. 1), so the restriction of $\Phi$ to $\mathfrak{h}$ is non-degenerate (Chap. VII, §1, no. 3, Prop. 10 (iii)).

(i) If $a$ is a nilpotent element of $\mathfrak{g}$, and if $\rho (a)$ is locally nilpotent,

for all $b\in \mathfrak{g}$.

(ii) If $\alpha \in R$ and if the images under $\rho$ of the elements of $\mathfrak{g}^{\alpha}$ and $\mathfrak{g}^{-\alpha}$ are locally nilpotent, the set of weights of V is stable under the reflection $s_{\alpha}$.

With the assumptions in (i), we have $\rho$((ad $a$)$^nb$) $=$ (ad $\rho (a)$)$^n\rho (b)$ for all $n\geq 0$, so $\rho (e^{ada}b) =e^{ad\rho(a)}\rho (b)$. On the other hand,

is assertion (ii) of Chap. VII, §3, no. 1, Lemma 1.

We now adopt the assumptions in (ii). Let $\theta_{\alpha}=e^{adX_{\alpha}}e^{adX_{-\alpha}}e^{adX_{\alpha}}$. By (i), there exists $S\in \mathbf{G}\mathbf{L}(V)$ such that $\rho (\theta_{\alpha}b) = S\rho (b)S^{-1}$ for all $b\in \mathfrak{g}$. Now $\theta_{\alpha}|\mathfrak{h}$ is the transpose of $s_{\alpha}($§2, no. 2, Lemma 1). Let $\lambda$ be a weight of V. There exists a non-zero element $x$ of V such that $\rho (h)x=\lambda (h)x$ for all $h\in \mathfrak{h}$. Then

for all $h\in \mathfrak{h}$. Consequently, $s_{\alpha}\lambda$ is a weight of V.

(i) There exists $S\in \mathbf{G}\mathbf{L}(V)$ such that $(s(x))_V= Sx_VS^{-1}$ for all $x\in \mathfrak{g}$.

(ii) If $s\in$ Aut$_e(\mathfrak{g})$, S can be chosen to be an element of $\mathbf{S}\mathbf{L}(V)$ leaving stable all the $\mathfrak{g}$-submodules of V.

Assertion (ii) follows from Lemma 1 (i). Now let $s\in$ Aut$_0(\mathfrak{g})$ and denote by $\rho$ the representation of $\mathfrak{g}$ defined by V. By (ii), the representations $\rho$ and $\rho \circ s$ become equivalent after extension of scalars. They are therefore equivalent (Chap. I, §3, no. 8, Prop. 13), hence the existence of S.

Indeed, $w$ is of the form $s^*$ with $s\in$ Aut$_e(\mathfrak{g},\mathfrak{h}) ($§2, no. 2, Cor. of Th. 2).

We denote by V a simple $\mathfrak{g}$-module and by $\mathscr{X}$ its set of weights.

a) Assume that V is finite dimensional. Then $\mathscr{X}$ is finite and non-empty (Prop. 1) and so has a maximal element $\omega$. Let $\alpha \in B$. Then $\omega +\alpha \notin \mathscr{X}$, which proves that $\omega$ is the highest weight of V (§6, no. 2, Lemma 2). On the other hand, there exists a homomorphism from $\mathfrak{s}\mathfrak{l}(2, k)$ to $\mathfrak{g}$ that takes $H$ to $H_{\alpha}$; by §1, Prop. 2 (i), $\omega (H_{\alpha})$ is an integer $\geq 0$, so $\omega \in P_{++}$.

b) Assume that V has a highest weight $\omega \in P_{++}$. Let $\alpha \in B$ and let $e$ be a primitive element of weight $\omega$ in V. Put $e_j=X_{-\alpha}^je$ for $j\geq 0$,

$m=\omega (H_{\alpha})\in \mathbf{N}$, and N = $\sum_{j=0}^mke_j$. By §1, no. 2, Prop. $1,X_{\alpha}e_{m+1}$ = 0.

If $\beta \in B$ and $\beta \not=\alpha$, then $[X_{\beta}, X_{-\alpha}] = 0$ so $X_{\beta}e_{m+1}=X_{\beta}X_{-\alpha}^{m+1}e$ = $X_{-\alpha}^{m+1}X_{\beta}e= 0$. If $e_{m+1}\not= 0$, we conclude that $e_{m+1}$ is primitive, which is absurd (§6, Prop. 3 (i)); so $e_{m+1}= 0$. Thus, by §1, no. 2, Cor. of Prop. 1, N is stable under the subalgebra $\mathfrak{s}_{\alpha}$ generated by $H_{\alpha}, X_{\alpha}$ and $X_{-\alpha}$. Now $\mathfrak{s}_{\alpha}$ is reductive in $\mathfrak{g}$, so the sum of the finite dimensional subspaces of V that are stable under $\mathfrak{s}_{\alpha}$ is a $\mathfrak{g}$-submodule of V (Chap. I, §6, no. 6, Prop. 7); since this sum is non-zero, it is equal to V. It follows from this that $(X_{\alpha})_V$ and $(X_{-\alpha})_V$ are locally nilpotent. In view of Lemma 1 (ii), $\mathscr{X}$ is stable under $s_{\alpha}$, and this holds for all $\alpha$. Hence $\mathscr{X}$ is stable under W. Now every orbit of W on P meets $P_{++}$ (Chap. VI, §1, no. 10). On the other hand, if $\lambda \in \mathscr{X}\cap P_{++}$, then $\lambda =\omega -\sum_{\alpha\in B}n_{\alpha}\alpha =\sum_{\alpha\in B}n'_{\alpha}\alpha$ with $n_{\alpha}\in \mathbf{N}$ and $n'_{\alpha}\geq 0$ for all $\alpha \in B$

(Chap. V, §3, no. 5, Lemma 6). So $\mathscr{X}\cap P_{++}$ is finite and hence so is $\mathscr{X}$. Since each weight has finite multiplicity (§6, no. 1, Prop. 1 (ii)), V is finite dimensional.

The $\mathfrak{g}$-modules $E(\lambda )$, where $\lambda$ is a fundamental weight, are called the fundamental $\mathfrak{g}$-modules; the corresponding representations are called the fundamental representations of $\mathfrak{g}$; they are absolutely irreducible (§6, no. 2, Prop. 3 (iv)).

If V is a finite dimensional $\mathfrak{g}$-module and $\lambda \in P_{++}$, the isotypical component of V of type $E(\lambda )$ is called the isotypical component of highest weight $\lambda$ of V.

(i) I = $-q, p$ and $q-p=\lambda (H_{\alpha})$.

(ii) For any integer $u\in 0, p+q,\lambda + (p-u)\alpha$ and $\lambda + (-q+u)\alpha$ are conjugate under $s_{\alpha}$, and $m_{-q+u}=m_{p-u}$.

(iii) If $t\in \mathbf{Z}$ and $t <(p-q)/2$, $(X_{\alpha})_V$ maps $V^{\lambda+t\alpha}$ injectively into $V^{\lambda+(t+1)\alpha}$.

(iv) The function $t \rightarrow m_t$ is increasing on $-q,(p-q)/2$ and decreasing on $(p-q)/2, p$.

Let $\alpha \in B$. Give V the $\mathfrak{s}\mathfrak{l}(2, k$)-module structure defined by the elements $X_{\alpha}, X_{-\alpha}, H_{\alpha}$ of $\mathfrak{g}$. Every non-zero element of $V^{\lambda+p\alpha}$ is then primitive. Consequently, $(\lambda +p\alpha )(H_{\alpha})\geq 0$ and $(X_{-\alpha})^rV^{\lambda+p\alpha}\not= 0$ for

(§1, no. 2, Prop. 2). It follows that $V^{\lambda+t\alpha}\not= 0$ for $p\geq t\geq p-(\lambda (H_{\alpha}) + 2p)$, so $p+q\geq \lambda (H_{\alpha}) + 2p$. Applying this result to $-\alpha$ gives

Hence $q-p=\lambda (H_{\alpha})$ and $\lambda +t\alpha \in \mathscr{X}$ for $p\geq t\geq -q$, which proves (i).

We have $s_{\alpha}(\alpha ) =-\alpha$, and $s_{\alpha}(\mu)\in \mu+k\alpha$ for all $\mu\in \mathfrak{h}^*$. Since W leaves $\mathscr{X}$ stable (Cor. 2 of Prop. $2$)$,s_{\alpha}$ leaves $\{\lambda -q\alpha , \lambda -q\alpha +\alpha , . . . , \lambda +p\alpha \}$ stable and takes $\lambda -q\alpha +u\alpha$ to $\lambda +p\alpha -u\alpha$ for all $u\in k$. Using Cor. 2 of Prop. 2 again, we see that $m_{-q+u}=m_{p-u}$ for every integer $u\in 0, p+q$. This proves (ii).

By §1, Cor. of Prop. $2, (X_{\alpha})_V|V^{\lambda+t\alpha}$ is injective for $t <(p-q)/2$. Now $(X_{\alpha})_V$ maps $V^{\lambda+t\alpha}$ to $V^{\lambda+(t+1)\alpha}$. Hence $m_{t+1}\geq m_t$ for $t <(p-q)/2$. Changing $\alpha$ to $-\alpha$, we see that $m_{t+1}\leq m_t$ for $t >(p-q)/2$. This proves (iii) and (iv).

This follows immediately from Prop. 3 (i).

Write $\nu =\sum_{\alpha\in B}c_{\alpha}.\alpha$, where $c_{\alpha}\in \mathbf{N}$ for all $\alpha \in B$. The corollary is clear

when $\sum_{\alpha\in B}c_{\alpha}= 0$; assume that $\sum_{\alpha\in B}c_{\alpha}>0$ and argue by induction on $\sum_{\alpha\in B}c_{\alpha}$. Let $(\cdot | \cdot )$ be a W-invariant non-degenerate positive symmetric bilinear form on $\mathfrak{h}^*_{\mathbf{R}}$. Then $(\nu |\sum_{\alpha\in B}c_{\alpha}.\alpha )>0$, so there exists $\beta \in B$ such that $c_{\beta}\geq 1$ and

$(\nu |\beta )>0$, hence $\nu (H_{\beta})\geq 1$. Since $\mu\in P_{++}$, it follows that $(\mu+\nu )(H_{\beta})\geq 1$. By Cor. $1,\mu+ (\nu -\beta )\in \mathscr{X}$, and it suffices to apply the induction hypothesis.

Replacing V by the $\mathfrak{g}$-submodule generated by $v$, if necessary, we can assume that V is simple. Let $\mathfrak{s}$ be the stabilizer. We have $(\mathfrak{n}_+)_Vv$ = 0, $(\mathfrak{h})_Vv\subset kv$. Let $\alpha \in B$ be such that $\omega (H_{\alpha}) = 0$. We have $\omega +\alpha \notin \mathscr{X}$, hence $\omega -\alpha \notin \mathscr{X}$ (Prop. 3 (i)) and consequently $(\mathfrak{g}^{-\alpha})_Vv= 0$. The preceding proves that $\mathfrak{p}_{\Sigma}\subset \mathfrak{s}$. If $\mathfrak{p}_{\Sigma}\not=\mathfrak{s}$, then $\mathfrak{s}=\mathfrak{p}_{\Sigma'}$, where $\Sigma '$ is a subset of B strictly containing $\Sigma$. Let $\beta \in \Sigma '$ -- $\Sigma$. Then $\mathfrak{g}^{-\beta}$ stabilizes $kv$, and hence annihilates $v$. But, since $\omega (H_{\beta})>0$, this contradicts Prop. 3 (iii). Q.E.D.

A subset $\mathscr{X}$ of P is called R-saturated if it satisfies the following condition: for all $\lambda \in \mathscr{X}$ and all $\alpha \in R$, we have $\lambda -t\alpha \in \mathscr{X}$ for all integers $t$ between 0 and $\lambda (H_{\alpha})$. Since $s_{\alpha}(\lambda ) =\lambda -\lambda (H_{\alpha})\alpha$, we see that an R-saturated subset of P is stable under W. Let $\mathscr{Y}\subset P$. An element $\lambda$ of $\mathscr{Y}$ is called R-extremal in $\mathscr{Y}$ if, for all $\alpha \in R$, either $\lambda +\alpha \notin \mathscr{Y}$ or $\lambda -\alpha \notin \mathscr{Y}$.

This follows immediately from Prop. 3.

(i) $\mathscr{X}$ is the smallest R-saturated subset of P containing $\omega$.

(ii) The R-extremal elements of $\mathscr{X}$ are the W-transforms of $\omega$.

(iii) If $\mu\in \mathscr{X}$, we have $\|\mu\| \leq \|\omega \|$. If, in addition, $\mu\not=\omega$, we have $\|\mu+\rho \|<\|\omega +\rho \|$. If $\mu$ is not R-extremal in $\mathscr{X}$, then $\|\mu\|<\|\omega \|$.

(iv) We have $\mathscr{X}= W.(\mathscr{X}\cap P_{++})$. An element $\lambda$ of $P_{++}$ belongs to $\mathscr{X}\cap P_{++}$ if and only if $\omega -\lambda \in Q_+$.

(i) Let $\mathscr{X}'$ be the smallest R-saturated subset of P containing $\omega$. We have $\mathscr{X}'\subset \mathscr{X}$ (Prop. 4). Assume that $\mathscr{X}\not=\mathscr{X}'$. Let $\lambda$ be a maximal element of $\mathscr{X}$ -- $\mathscr{X}'$. Since $\lambda \not=\omega$, there exists $\alpha \in B$ such that $\lambda +\alpha \in \mathscr{X}$. Introduce $p$ and $q$ as in Prop. 3. Since $\lambda$ is maximal in $\mathscr{X}$ -- $\mathscr{X}',\lambda +p\alpha \in \mathscr{X}'$. By Prop. 3 (ii), $\lambda -q\alpha \in \mathscr{X}'$ since $\mathscr{X}'$ is stable under W. Hence $\lambda +u\alpha \in \mathscr{X}'$ for every integer $u$ in the interval $-q, p$. This contradicts $\lambda \notin \mathscr{X}'$ and proves (i).

(ii) It is clear that $\omega$ is an R-extremal element of $\mathscr{X}$; its W-transforms are therefore also R-extremal in $\mathscr{X}$. Let $\lambda$ be an R-extremal element of $\mathscr{X}$; we shall prove that $\lambda \in W.\omega$. Since there exists $w\in W$ such that $w\lambda \in P_{++}$ (Chap. VI, §1, no. 10), we can assume that $\lambda \in P_{++}$. Let $\alpha \in B$; introduce $p$ and $q$ as in Prop. 3. Since $\lambda$ is R-extremal, either $p= 0$ or $q= 0$. Since

we cannot have $p >0$. Hence $p= 0$ and $\lambda =\omega$.

(iii) Let $\mu\in \mathscr{X}\cap P_{++}$. Then $\omega +\mu\in P_{++}$ and $\omega -\mu\in Q_+($§6, no. 1, Prop. 1), so $0\leq (\omega -\mu|\omega +\mu) = (\omega |\omega )-(\mu|\mu)$; hence, $(\mu|\mu)\leq (\omega |\omega )$, and this extends to all $\mu\in \mathscr{X}$ by using the Weyl group. If $\mu\in \mathscr{X}$ -- $\{\omega \}$,

Now $\omega -\mu=\sum_{\alpha\in B}n_{\alpha}\alpha$ with integers $n_{\alpha}\geq 0$ not all zero, so $(\omega -\mu|\rho )>0$ since $(\rho |\alpha )>0$ for all $\alpha \in B$ (Chap. VI, §1, no. 10, Prop. 29 (iii)). If $\mu$ is not R-extremal in $\mathscr{X}$, there exists $\alpha \in R$ such that $\mu+\alpha \in \mathscr{X}$ and $\mu-\alpha \in \mathscr{X}$; then

$\|\mu\|<$ sup($\|\mu+\alpha \|,\|\mu-\alpha \|$)$\leq$ sup$_{\lambda\in\mathscr{X}}\|\lambda \|$

and this last upper bound is $\|\omega \|$ by the preceding.

(iv) We have $\mathscr{X}= W.(\mathscr{X}\cap P_{++})$ by Chap. VI, §1, no. 10. If $\lambda \in \mathscr{X}$, then $\omega -\lambda \in Q_+($§6, no. 1, Prop. 1). If $\lambda \in P_{++}$ and $\omega -\lambda \in Q_+$, then $\lambda \in \mathscr{X}$ (Cor. 2 of Prop. 3).

Since $\mathscr{X}$ is stable under $W,\mathscr{X}$ is the smallest R-saturated set containing $\mathscr{X}\cap P_{++}$. By Prop. 5 (i), $\mathscr{X}$ is the set of weights of $\bigoplus_{\lambda\in\mathscr{X}\cap P_{++}}E(\lambda )$.

(i) $\mathscr{X}= W.\lambda$;

(ii) all the elements of $\mathscr{X}$ have the same norm;

(iii) for all $\alpha \in R$, we have $\lambda (H_{\alpha})\in \{0,1,-1\}$.

Every non-empty R-saturated subset of P contains an element $\lambda$ satisfying the above conditions.

Introduce the condition:

(ii$'$) for all $\alpha \in R$ and for every integer $t$ between 0 and $\lambda (H_{\alpha})$,

(i) $=\Rightarrow$ (ii) $=\Rightarrow$ (ii$'$)$:$ This is clear.

(ii$'$) $=\Rightarrow$ (iii): Assume that condition (ii$'$) is satisfied. Let $\alpha \in R$. We have $\|\lambda \|=\|\lambda -\lambda (H_{\alpha})\alpha \|$, so $\|\lambda -t\alpha \|<\|\lambda \|$ for every integer $t$ strictly between 0 and $\lambda (H_{\alpha})$; hence, there can be no such integers, so $|\lambda (H_{\alpha})| \leq 1$.

(iii) $=\Rightarrow$ (i): Assume that condition (iii) is satisfied. Let $w\in W$ and $\alpha \in R$. Then $(w\lambda )(H_{\alpha}) =\lambda (H_{w^{-1}\alpha})\in \{0,1,-1\}$; thus, if $t$ is an integer between 0 and $(w\lambda )(H_{\alpha}),w\lambda -t\alpha$ is equal to $w\lambda$ or $s_{\alpha}(w\lambda )$. This proves that $W.\lambda$ is R-saturated, so $\mathscr{X}= W.\lambda$.

Let $\mathscr{Y}$ be a non-empty R-saturated subset of P. There exists in $\mathscr{Y}$ an element $\lambda$ of minimum norm. It is clear that $\lambda$ satisfies condition (ii$'$), hence the last assertion of the proposition.

(iv) for all $\alpha \in R$ and all $x\in \mathfrak{g}^{\alpha}$, we have $(x_V)^2= 0$.

If these conditions are satisfied, all the weights of V have multiplicity 1.

If (i) is satisfied, then $\mathscr{X}= W.\lambda$ and the weights all have the same multiplicity as $\lambda$ (Cor. 2 of Prop. 2), in other words, multiplicity 1. Moreover, if $w\in W$ and $\alpha \in R,w\lambda +t\alpha$ cannot be a weight of V unless $|t| \leq 1$; thus, if $x\in \mathfrak{g}^{\alpha}$,

so $(x_V)^2= 0$, which proves that (i) $=\Rightarrow$ (iv).

Conversely, assume that (iv) is satisfied. Let $\alpha \in R$, and give V the $\mathfrak{s}\mathfrak{l}(2, k)$-module structure defined by the elements $X_{\alpha}, X_{-\alpha}, H_{\alpha}$ of $\mathfrak{g}$. Condition (iv), applied to $x=X_{\alpha}$, implies that the weights of the $\mathfrak{s}\mathfrak{l}(2, k$)-module V belong to $\{0,1,-1\}$ (cf. §1, no. 2, Cor. of Prop. 2). In particular, $\lambda (H_{\alpha})\in \{0,1,-1\}$, so (iv) $=\Rightarrow$ (iii).

(v) $\lambda (H) = 1$;

(vi) there exists $i\in J$ such that $\lambda =\varpi_i$.

The $\varpi_i$, for $i\in J$, form a system of representatives in P(R) of the non-zero elements of $P(R)/Q(R)$.

Let $\lambda =u_1\varpi_1+\cdots +u_l\varpi_l$, where $u_1, . . . , u_l$ are integers $\geq 0$ and not all zero. Then $\lambda (H) =u_1n_1+\cdots +u_ln_l$ and $n_1\geq 1, . . . , n_l\geq 1$, which gives the equivalence of (v) and (vi) immediately. On the other hand, $\lambda (H) =$ sup$_{\alpha\in R_+}\lambda (H_{\alpha})$, and $\lambda (H)>0$ since $\lambda$ is a non-zero element of $P_{++}$. Hence

condition (v) is equivalent to the condition $\lambda (H_{\alpha})\in \{0,1\}$ for all $\alpha \in R$, in other words to condition (iii) of Prop. 6.

The last assertion of the proposition follows from Chap. VI, §2, no. 3, Cor. of Prop. 6.

(i) The component of $E\otimes F$ of highest weight $\lambda +\mu$ is a simple submodule, generated by $(E\otimes F)^{\lambda+\mu}= E^{\lambda}\otimes F^\mu$.

(ii) The highest weight of any simple submodule of $E\otimes F$ is $\leq \lambda +\mu$ (cf. §9, Prop. 2).

If $\alpha , \beta \in P$ and if $E^{\alpha}\otimes F^{\beta}\not= 0$, then $\alpha \leq \lambda$ and $\beta \leq \mu$. Consequently, $(E\otimes F)^{\lambda+\mu}$ is equal to $E^{\lambda}\otimes F^\mu$, and hence is of dimension 1, and $\lambda +\mu$ is the highest weight of $E\otimes F$. Every non-zero element of $E^{\lambda}\otimes F^\mu$ is primitive. By Prop. 4 of §6, no. 2, the length of the isotypical component of $E\otimes F$ of highest weight $\lambda +\mu$ is 1.