Exercise 18, § 8

Let $\mathfrak{a}$ be a semi-simple Lie algebra, $l=$ rk($\mathfrak{a}$)$, I$ the algebra of invariant polynomial functions on $\mathfrak{a}$, and $P_1, . . . ,P_l$ homogeneous elements of I generating I as an algebra. The $P_i$ define a polynomial map $P :\mathfrak{a}\rightarrow k^l$. If $x\in \mathfrak{a}$, denote by $D_xP :\mathfrak{a}\rightarrow k^l$ the tangent linear map of P at $x$ (Chap. VII, App. I, no. 2).

a) Let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{a}$, and let $x\in \mathfrak{h}$. Prove the equivalence of:

(i) $D_xP|\mathfrak{h}$ is an isomorphism from $\mathfrak{h}$ to $k^l$;

(ii) $x$ is regular.

(Reduce to the split case. Choose a basis of $\mathfrak{h}$, and denote by $d(x)$ the determinant of the matrix of $D_xP|\mathfrak{h}$ relative to this basis. Show, by means of Prop. 5 of Chap. V, §5, no. 4, that there exists $c\in k^*$ such that $d(x)^2=c\prod_{\alpha\in R}\alpha (x)$,

where $\alpha$ belongs to the set R of roots of $(\mathfrak{a},\mathfrak{h}).)$

If these conditions are satisfied, $\mathfrak{a}=\mathfrak{h}\oplus$Im ad($x$), and Ker $D_xP =$ Im ad($x$) (use Exerc. 6 to show that $D_xP$ vanishes on Im ad($x$)).

b) Show that the set of $x\in \mathfrak{a}$ such that $D_xP$ is of rank $l$ is a dense open subset of $\mathfrak{a}$ in the Zariski topology.