Exercise 26, § 7

Apply the notations and results of Exerc. 25 to the case in which $\mathfrak{a}=\mathfrak{g}$ (split case).

a) Let $x\in G$ and let $\sigma =v(x)$ be its image in Aut($\mathfrak{g}$). If $\rho$ is a representation of $\mathfrak{g},\rho$ and $\rho \circ \sigma$ are equivalent. Deduce (cf. no. 2, Remark 1) that $\sigma$ belongs to Aut$_0(\mathfrak{g})$. Extend this result to arbitrary semi-simple algebras.

b) Let $\varphi \in T_P=$ Hom(P$, k^*$), where P = P(R). If E is a $\mathfrak{g}$-module, let $\varphi_E$ be the endomorphism of E whose restriction to each $E^{\lambda}(\lambda \in P)$ is the homothety of ratio $\varphi (\lambda )$. Show that there exists a unique element $t(\varphi )\in G$ such that $t(\varphi )_E=\varphi_E$ for all E (Use Exerc. $24c$), and the characterizations (ii) and (vi) of Exerc. 25.) This gives a homomorphism $t: T_P\rightarrow G$. Show that $t$ is injective. We use this to identify $T_P$ with a subgroup of G. The composite $T_P\rightarrow G\rightarrow$ Aut($\mathfrak{g}$) is the homomorphism denoted by $f\circ q$ in §5, no. 2.

c) Let $x\in G$ be such that $\sigma =v(x)$ belongs to the subgroup $f(T_Q)$ of Aut$_0(\mathfrak{g})$ (§5, no. 2), in other words, such that it operates trivially on $\mathfrak{h}$; denote the element of $T_Q=$ Hom(Q$, k^*$) corresponding to $x$ by $\psi$. We are going to show that $x$ belongs to $T_P$. Prove successively:

$c_1)$ If E is a $\mathfrak{g}$-module, $x_E$ is an $\mathfrak{h}$-endomorphism of E.

(Use the $\mathfrak{g}$-homomorphism $\mathfrak{g}\otimes E\rightarrow E$, and the fact that $x$ operates trivially on $\mathfrak{h}.$) In particular, the $E^\mu$ are stable under $x_E$.

$c_2)$ There exists $\varphi \in T_P$ such that, for every $\mathfrak{g}$-module E, and every primitive element $e$ of E of weight $\lambda$, we have $x_Ee=\varphi (\lambda )e$.

(Choose $\varphi$ such that this relation is true when E is a fundamental module $E(\varpi_{\alpha})$. Deduce the case of the $E(\lambda ),\lambda \in P_{++}$, by using the embedding of

$^{16*}$In this case, it can be shown that $U'$ is the bigebra of the simply-connected

semi-simple algebraic group A with Lie algebra $\mathfrak{a}$, and that G is the group of

$k$-points of $A$.\* such a module in a tensor product of the $E(\varpi_{\alpha})$. Pass from this to the general case.)

$c_3)$ Choose $\varphi$ as in $c_2)$. Let E be a simple $\mathfrak{g}$-module of highest weight $\lambda$, and let $\mu$ be a weight of E; then $\lambda -\mu\in Q$. Show that the restriction of $x_E$ to $E^\mu$ is the homothety of ratio $\varphi (\lambda )\psi (\mu-\lambda )$. (Same method as for $c_1$)$.)$

$c_4)$ If $\lambda , \mu\in P_{++}$, and if $\alpha \in B$ is not orthogonal to either $\lambda$ or $\mu$, then

so $\varphi (\alpha ) =\psi (\alpha )$. (Use $c_2$)$,c_3)$, and the embedding of $E(\lambda +\mu-\alpha )$ in $E(\lambda )\otimes E(\mu)$, cf. Exerc. 17.)

$c_5)$ Deduce from $c_4)$ that $\varphi |Q =\psi$, and use $c_3)$ to deduce that $x=t(\varphi )$. d) Identify $T_Q$ with a subgroup of Aut$_0(\mathfrak{g})$ by means of $f$. By a), we have

Aut$_e(\mathfrak{g})\subset v(G)\subset$ Aut$_0(\mathfrak{g})$

and, by $c),v(G)\cap T_Q=$ Im(T$_P$). Deduce (cf. §5, no. 3) that Aut$_e(\mathfrak{g})\cap T_Q=$ Im(T$_P$) and that $f(G) =$ Aut$_e(\mathfrak{g})$. The canonical map

$\iota : T_Q/$Im(T$_P$)$\rightarrow$ Aut$_0(\mathfrak{g})/$Aut$_e(\mathfrak{g})$

is therefore an isomorphism.

e) The kernel of $v: G\rightarrow$ Aut$_e(\mathfrak{g})$ is equal to the kernel of $T_P\rightarrow T_Q$; it is isomorphic to

Hom(P$/Q, k^*$);

this is a finite abelian group contained in the centre of G, and its order divides (P : Q); if $k$ is algebraically closed, it is isomorphic to the dual of $P/Q ($Algebra, Chap. VII, §4, no. 8).

f) Let $\alpha \in R, X_{\alpha}\in \mathfrak{g}^{\alpha}$ and $X_{-\alpha}\in \mathfrak{g}^{-\alpha}$ be such that $[X_{\alpha}, X_{-\alpha}] =-H_{\alpha}$, and let $\rho_{\alpha}$ be the corresponding representation of $\mathfrak{s}\mathfrak{l}(2, k)$ on $\mathfrak{g}$. If E is a $\mathfrak{g}$-module, deduce (§1, no. 4) a representation of $\mathbf{S}\mathbf{L}(2, k)$ on E, and hence (Exerc. 25 $b$)$,c))$ a homomorphism

Show that Im($\varphi_{\alpha}$) contains the elements of $T_P$ of the form $\lambda \rightarrow t^{\lambda(H_{\alpha})},t\in k^*$. Deduce that the Im($\varphi_{\alpha}$)$,\alpha \in B$, generate G (show first that the group they generate contains $T_P$). In particular, G is generated by the $e^n$, with $n\in \mathfrak{g}^{\alpha}$, $\alpha \in B\cup -B$. The derived group of G is equal to G.

g) If a subgroup $G'$ of G is such that $v(G') =$ Aut$_e(\mathfrak{g})$, then $G'= G$ (use $f$)$)$. h) Let E be a faithful $\mathfrak{g}$-module, and let $\Gamma$ be the subgroup of P generated by the weights of E. Then $P\supset \Gamma \supset Q$, cf. Exerc. 5. Show that the kernel of the canonical homomorphism $G\rightarrow \mathbf{G}\mathbf{L}(E)$ is equal to the subgroup of $T_P$ consisting of the elements $\varphi$ whose restriction to $\Gamma$ is trivial. In particular, if $\Gamma$ = P the homomorphism $G\rightarrow \mathbf{G}\mathbf{L}(E)$ is injective. If $\Gamma$ = Q, this homomorphism factorizes as G $\longrightarrow^v$ Aut$_e(\mathfrak{g})\longrightarrow \mathbf{G}\mathbf{L}(E)$, and the homomorphism Aut$_e(\mathfrak{g})\rightarrow \mathbf{G}\mathbf{L}(E)$ is injective.