Exercise 10, § 3

Let $\mathfrak{a}$ be a semi-simple Lie algebra, and $x$ an element of $\mathfrak{a}$. We say that $x$ is diagonalizable if ad $x$ is diagonalizable (Algebra, Chap. VII), in other words, if there exists a basis of $\mathfrak{a}$ consisting of eigenvectors of ad $x$.

a) Let $\mathfrak{c}$ be a commutative subalgebra of $\mathfrak{a}$ consisting of diagonalizable elements, and let L be the set of weights of $\mathfrak{c}$ in the representation ad$_{\mathfrak{a}}$. The set L is a finite subset of $\mathfrak{c}^*$ containing 0 (unless $\mathfrak{a}= 0$) and $\mathfrak{a}=\bigoplus_{\lambda\in L}\mathfrak{a}^{\lambda}(\mathfrak{c})$. Show that there exists a subset M of L such that L -- $\{0\}$ is the disjoint union of M and $-M$, and such that $(M + M)\cap L\subset M$. If M has these properties, put $\mathfrak{a}^M=\bigoplus_{\lambda\in M}\mathfrak{a}^{\lambda}(\mathfrak{c})$, and $\mathfrak{p}^M=\mathfrak{a}^0(\mathfrak{c})\oplus \mathfrak{a}^M$. The algebra $\mathfrak{a}^0(\mathfrak{c})$ is the commutant of $\mathfrak{c}$ in $\mathfrak{a}$; it is reductive in $\mathfrak{a}$ (Chap. VII, §1, no. 5), and its Cartan subalgebras are the Cartan subalgebras of $\mathfrak{a}$ (Chap. VII, §2, no. 3). Show that $\mathfrak{p}^M$ is a

$^{11}$This exercise was communicated to us by A. JOSEPH. parabolic subalgebra of $\mathfrak{a}$, and that $\mathfrak{a}^M$ is the set of nilpotent elements of the radical of $\mathfrak{p}^M$. (Use the fact that $\mathfrak{c}$ is contained in a Cartan subalgebra of $\mathfrak{a}$ and reduce, by extension of scalars, to the case in which this subalgebra is splitting.)

b) Retain the hypotheses and notations of a), and assume in addition that $\mathfrak{a}$ is splittable. Show that $\mathfrak{c}$ is contained in a splitting Cartan subalgebra of $\mathfrak{a}$. (Take a splitting Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{a}$ contained in $\mathfrak{p}^M$; then there exists a unique Cartan subalgebra $\mathfrak{h}'$ of $\mathfrak{a}^0(\mathfrak{c})$ such that $\mathfrak{h}$ is contained in $\mathfrak{h}'\oplus \mathfrak{a}^M$; show that $\mathfrak{h}'$ is a splitting Cartan subalgebra of $\mathfrak{a}$, and that $\mathfrak{h}'$ contains $\mathfrak{c}.$)