Exercise 15, § 8

Let $x$ be a diagonalizable element of $\mathfrak{g}($§3, Exerc. 10), and $y$ a semi-simple element of $\mathfrak{g}$ such that $f(x) =f(y)$ for every invariant polynomial function $f$ on $\mathfrak{g}$. Show that there exists $s\in$ Aut$_e(\mathfrak{g})$ such that $sy=x$. (Remark that ad $x$ and ad $y$ have the same characteristic polynomial, cf. Exerc. 8, hence

$^{17}$For more details on Exercises 10 to 13, see: I. N. BERNSTEIN, I. M. GELFAND

and S. I. GELFAND, Structure of representations generated by highest weight

vectors [in Russian], Funct. Anal. i evo prilojenie, Vol. V (1971), pp. 1-9.

In this memoir, it is further proved that, if $\lambda , \lambda '\in \mathfrak{h}^*$ are such that $Z(\lambda -\rho )\subset$

$Z(\lambda '-\rho )$, there exist $\gamma_1, . . . , \gamma_n\in R_+$ such that $\lambda$ = $s_{\gamma_n}. . . s_{\gamma_2}s_{\gamma_1}\lambda '$ and

$(s_{\gamma_i}. . . s_{\gamma_1}\lambda ')(H_{\gamma_{i+1}})\in \mathbf{N}$ for $0\leq i\leq n$. It follows that $Z(\lambda -\rho )$ is simple if

and only if $\lambda (H_{\alpha})\in \mathbf{N}^*$ for all $\alpha \in R_+$. the fact that $y$ is diagonalizable. Next, reduce to the case in which $x$ and $y$ are contained in $\mathfrak{h}$ by using Exerc. 10 of §3, and use Th. 1 (i) and Lemma 6 to prove that $x$ and $y$ are conjugate under W.)