Exercise 8, § 3

Let $G$ be a finite-dimensional connected real Lie group and $M$ a finite-dimensional connected manifold of class $C^\infty$; let there be given a law of right operation of class $C^\infty$ of $G$ on $M$. For all $x \in M$, let $\rho(x)$ be the corresponding orbital mapping.

(a) For $G$ to operate transitively on $M$, it is necessary and sufficient that, for all $x \in M$, the mapping $T_e(\rho(x))$ of $L(G)$ into $T_x(M)$ be surjective. (To see that the condition is sufficient, observe that then every $G$-orbit in $M$ is open.)

(b) Suppose that $G$ operates transitively on $M$. We identify $M$ with a homogeneous space $H|G$, where $H$ is the stabilizer of a point $x_0$ of $M$. The action of $G$ on $M$ is called imprimitive if there exists a closed submanifold $V$ of $M$ of class $C^\infty$ such that $0 < \dim V < \dim M$ and for which every transform $V_s$ (where $s \in G$) is either equal to $V$ or disjoint from $V$. For this to be so, it is necessary and sufficient that there exist a Lie subgroup $L$ of $G$ such that $H \subset L \subset G$ and $\dim(H) < \dim(L) < \dim(G)$. If this is not so, the action of $G$ on $M$ is called primitive; for this it suffices that there exist no subalgebra of $L(G)$ lying between $L(G)$ and $L(H)$ and distinct from $L(G)$ and $L(H)$.

(c) Under the hypothesis of (b), let $\mathscr{L}$ be the image of $L(G)$ under the law of infinitesimal operation associated with the action of $G$. Let $\mathscr{E}$ be the algebra of functions of class $C^\infty$ on $M$ and $\mathfrak{m}$ the maximal ideal of $\mathscr{E}$ consisting of the functions in $\mathscr{E}$ which are zero at $x_0$; we denote by $\mathscr{L}_p$ for $p = -1, 0, 1, 2, \ldots$ the set of $x \in \mathscr{L}$ such that $\theta(X)f \in \mathfrak{m}^{p+1}$ for all $f \in \mathscr{E}$; then $\mathscr{L}_{-1} = \mathscr{L}$ and $\mathscr{L}_0$ is the image of $L(H)$ under the law of infinitesimal operation. If $(U, \phi, \mathbf{R}^n)$ is a chart of $M$ centered on $x_0$ and $X_1, \ldots, X_n$ are the vector fields on $U$ defined by the chart, the elements of $\mathscr{L}_p$ are the $a_1X_1 + \cdots + a_nX_n$ with $a_1, \ldots, a_n$ in $\mathfrak{m}^p|U$. Show that $[\mathscr{L}_p, \mathscr{L}_q] \subset \mathscr{L}_{p+q}$ (with the convention $\mathscr{L}_{-2} = \mathscr{L}$). If there exists $Y \in \mathscr{L}_p$ (for $p \geq 0$) such that $Y \notin \mathscr{L}_{p+1}$, there exists $x \in L$ such that $[Y, X] \notin \mathscr{L}_p$. The $\mathscr{L}_p$ are Lie subalgebras of $\mathscr{L}$. For $p \geq 0$, $\mathscr{L}_p$ is an ideal of $\mathscr{L}_0$. Let $\rho$ be the canonical linear representation of $H$ on $T_{x_0}(M)$. The Lie algebra of $H/(Ker \rho)$ is isomorphic to $\mathscr{L}_0/\mathscr{L}_1$.

(d) Suppose further that the intersection of the $\mathscr{L}_p$ is $\{0\}$. Then there exists a greatest index $r$ such that $\mathscr{L}_r \neq \{0\}$ and all the $\mathscr{L}_j$ of indices $\leq r$ are distinct. For $p \geq 0$,

(e) The hypothesis of (d) is satisfied if M is analytic and G operates analytically on M.