Exercise 17, § 11

Let $(x, h, y)$ be a principal $\mathfrak{s}\mathfrak{l}_2$-triplet in the semi-simple algebra $\mathfrak{g}$. For all $i\in \mathbf{Z}$, denote by $\mathfrak{g}_i$ the eigenspace of ad $h$ relative to the eigenvalue $i$; we have $\mathfrak{g}=\bigoplus_{i\in\mathbf{Z}}\mathfrak{g}_i$, and $\mathfrak{g}_i= 0$ if $i$ is odd. The direct sum $\mathfrak{b}$ of the $\mathfrak{g}_i,i\geq 0$, is a

Borel subalgebra of $\mathfrak{g}$, and $\mathfrak{g}_0$ is a Cartan subalgebra.

a) Show that, for all $z\in \mathfrak{b}$, the commutant of $y+z$ in $\mathfrak{g}$ is of dimension $l=$ rk($\mathfrak{g}$).

b) Let I be the algebra of invariant polynomial functions on $\mathfrak{g}$, and $P_1, . . . ,P_l$ homogeneous elements of I generating I; put deg(P$_i$) $=k_i=m_i+ 1$. Show that the commutant $\mathfrak{c}$ of $x$ in $\mathfrak{g}$ has a basis $x_1, . . . , x_l$ with $x_i\in \mathfrak{g}_{2m_i}$ (cf. Exerc. 11).

c) Let $i\in 1, l$, and let $J_i$ (resp. $K_i$) be the set of $j\in (1, l)$ such that $m_j=m_i$ (resp. $m_j< m_i$). Let $f_i\in k[X_1, . . . ,X_l]$ be the polynomial such that

$f_i(a_1, . . . , a_l) = P_i(y+\sum_{j=1}^{j=l}a_jx_j)$ for $(a_j)\in k^l$.

Show that $f_i$ is the sum of a linear form $L_i$ in the $X_j,j\in J_i$, and a polynomial in the $X_j,j\in K_i$. (If $t\in k^*$, the automorphism of $\mathfrak{g}$ equal to $t^i$ on $\mathfrak{g}_i$ belongs to Aut$_0(\mathfrak{g})$ and takes $y$ to $t^{-2}y$ and $x_j$ to $t^{2m_j}x_j$. Use the invariance of $P_i$ under this automorphism.)

d) Let P be the map from $\mathfrak{g}$ to $k^l$ defined by the $P_i$. If $z\in \mathfrak{g}$, denote by $D_zP :\mathfrak{g}\rightarrow k^l$ the tangent linear map of P at $z$ (Chap. VII, App. I, no. 2).

Show that $\mathfrak{c}\cap$ Im ad($y-x$) $= 0$ (decompose $\mathfrak{g}$ into a direct sum of simple submodules relative to the subalgebra generated by the given $\mathfrak{s}\mathfrak{l}_2$-triplet). Deduce that the restriction of $D_{y-x}P$ to $\mathfrak{c}$ is an isomorphism from $\mathfrak{c}$ to $k^l$ (use the preceding exercise and Exerc. $18a$) of §8). Show, by using this result, that the determinant of the linear forms $L_i$ defined in c) is $\not= 0$, and deduce the following results:

$d_1)$ the polynomials $f_1, . . . , f_l$ are algebraically independent and generate $k[X_1, . . . ,X_l]$;

$d_2)$ the map $z \rightarrow P(y+z)$ from $\mathfrak{c}$ to $k^l$ is bijective polynomial, and the inverse map is polynomial;

$d_3)$ for all $z\in y+\mathfrak{c}$, the linear map $D_zP|\mathfrak{c}$ is of rank $l$.

In particular, the map $P :\mathfrak{g}\rightarrow k^l$ is surjective.

e) If $k$ is algebraically closed, every element of $\mathfrak{g}$ whose commutant is of dimension $l$ is conjugate under Aut$_0(\mathfrak{g})$ to a unique element of $y+\mathfrak{c}$. f) Give an example of a simple Lie algebra with no principal $\mathfrak{s}\mathfrak{l}_2$-triplet, for which the map P is not surjective (take $k=\mathbf{R}$ and $l= 1$).